diff --git a/benchmark/ec/ec_double_benchmark_gpu.cc b/benchmark/ec/ec_double_benchmark_gpu.cc
index c1bbe8bf2..960eea6c4 100644
--- a/benchmark/ec/ec_double_benchmark_gpu.cc
+++ b/benchmark/ec/ec_double_benchmark_gpu.cc
@@ -9,9 +9,9 @@
 #include "tachyon/base/time/time_interval.h"
 #include "tachyon/device/gpu/gpu_memory.h"
 #include "tachyon/math/elliptic_curves/bn/bn254/g1_gpu.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
 #include "tachyon/math/elliptic_curves/short_weierstrass/kernels/elliptic_curve_ops.cu.h"
 #include "tachyon/math/elliptic_curves/test/random.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon {
 
diff --git a/tachyon/c/math/elliptic_curves/msm/msm_gpu.h b/tachyon/c/math/elliptic_curves/msm/msm_gpu.h
index b48bc6cfd..8cd94e1e8 100644
--- a/tachyon/c/math/elliptic_curves/msm/msm_gpu.h
+++ b/tachyon/c/math/elliptic_curves/msm/msm_gpu.h
@@ -13,9 +13,9 @@
 #include "tachyon/c/math/elliptic_curves/msm/msm_input_provider.h"
 #include "tachyon/device/gpu/scoped_mem_pool.h"
 #include "tachyon/device/gpu/scoped_stream.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
 #include "tachyon/math/elliptic_curves/msm/variable_base_msm_gpu.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon::c::math {
 
diff --git a/tachyon/c/zk/plonk/halo2/BUILD.bazel b/tachyon/c/zk/plonk/halo2/BUILD.bazel
index 1379032f1..12d7741e9 100644
--- a/tachyon/c/zk/plonk/halo2/BUILD.bazel
+++ b/tachyon/c/zk/plonk/halo2/BUILD.bazel
@@ -180,10 +180,10 @@ tachyon_cc_library(
         "//tachyon/base/buffer:endian_auto_reset",
         "//tachyon/base/buffer:read_only_buffer",
         "//tachyon/base/containers:container_util",
-        "//tachyon/math/elliptic_curves:points",
         "//tachyon/math/finite_fields:cubic_extension_field",
         "//tachyon/math/finite_fields:prime_field_base",
         "//tachyon/math/finite_fields:quadratic_extension_field",
+        "//tachyon/math/geometry:affine_point",
         "//tachyon/math/polynomials/univariate:univariate_evaluations",
         "//tachyon/math/polynomials/univariate:univariate_polynomial",
         "//tachyon/zk/expressions:expression_factory",
diff --git a/tachyon/c/zk/plonk/halo2/buffer_reader.h b/tachyon/c/zk/plonk/halo2/buffer_reader.h
index b1768a12c..25ccc9eaf 100644
--- a/tachyon/c/zk/plonk/halo2/buffer_reader.h
+++ b/tachyon/c/zk/plonk/halo2/buffer_reader.h
@@ -15,10 +15,10 @@
 #include "tachyon/base/logging.h"
 #include "tachyon/c/zk/plonk/halo2/bn254_gwc_pcs.h"
 #include "tachyon/c/zk/plonk/halo2/bn254_shplonk_pcs.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
 #include "tachyon/math/finite_fields/cubic_extension_field.h"
 #include "tachyon/math/finite_fields/prime_field_base.h"
 #include "tachyon/math/finite_fields/quadratic_extension_field.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/math/polynomials/univariate/univariate_evaluations.h"
 #include "tachyon/math/polynomials/univariate/univariate_polynomial.h"
 #include "tachyon/zk/expressions/expression_factory.h"
diff --git a/tachyon/crypto/commitments/fri/fri.h b/tachyon/crypto/commitments/fri/fri.h
index 6524970ce..afd84abcd 100644
--- a/tachyon/crypto/commitments/fri/fri.h
+++ b/tachyon/crypto/commitments/fri/fri.h
@@ -69,7 +69,7 @@ class FRI final
         // Pᵢ₊₁(X) = Pᵢ_even(X²) + β * Pᵢ_odd(X²)
         beta = writer->SqueezeChallenge();
         VLOG(2) << "FRI(beta[" << i - 1 << "]): " << beta.ToHexString(true);
-        folded_poly = cur_poly->template Fold<false>(beta);
+        folded_poly = cur_poly->Fold(beta);
         BinaryMerkleTree<F, F, MaxDegree + 1> tree(storage_->GetLayer(i),
                                                    hasher_);
         evals = sub_domains_[i - 1]->FFT(folded_poly);
@@ -82,7 +82,7 @@ class FRI final
     beta = writer->SqueezeChallenge();
     VLOG(2) << "FRI(beta[" << num_layers - 1
             << "]): " << beta.ToHexString(true);
-    folded_poly = cur_poly->template Fold<false>(beta);
+    folded_poly = cur_poly->Fold(beta);
     return writer->WriteToProof(folded_poly[0]);
   }
 
diff --git a/tachyon/crypto/commitments/kzg/kzg.h b/tachyon/crypto/commitments/kzg/kzg.h
index 934f658f9..76abfe10a 100644
--- a/tachyon/crypto/commitments/kzg/kzg.h
+++ b/tachyon/crypto/commitments/kzg/kzg.h
@@ -19,7 +19,7 @@
 #include "tachyon/base/logging.h"
 #include "tachyon/crypto/commitments/batch_commitment_state.h"
 #include "tachyon/math/elliptic_curves/msm/variable_base_msm.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/point_conversions.h"
 #include "tachyon/math/polynomials/univariate/univariate_evaluation_domain.h"
 
 #if TACHYON_CUDA
diff --git a/tachyon/crypto/commitments/pedersen/pedersen.h b/tachyon/crypto/commitments/pedersen/pedersen.h
index 17e41637f..ecfec9cd7 100644
--- a/tachyon/crypto/commitments/pedersen/pedersen.h
+++ b/tachyon/crypto/commitments/pedersen/pedersen.h
@@ -21,7 +21,7 @@
 #include "tachyon/base/strings/string_util.h"
 #include "tachyon/crypto/commitments/vector_commitment_scheme.h"
 #include "tachyon/math/elliptic_curves/msm/variable_base_msm.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 #if TACHYON_CUDA
 #include "tachyon/device/gpu/scoped_mem_pool.h"
diff --git a/tachyon/crypto/transcripts/BUILD.bazel b/tachyon/crypto/transcripts/BUILD.bazel
index e117fc9e3..3f6c961f2 100644
--- a/tachyon/crypto/transcripts/BUILD.bazel
+++ b/tachyon/crypto/transcripts/BUILD.bazel
@@ -22,7 +22,7 @@ tachyon_cc_library(
     name = "transcript_traits",
     hdrs = ["transcript_traits.h"],
     deps = [
-        "//tachyon/math/elliptic_curves:points",
         "//tachyon/math/finite_fields:prime_field_base",
+        "//tachyon/math/geometry:affine_point",
     ],
 )
diff --git a/tachyon/crypto/transcripts/transcript_traits.h b/tachyon/crypto/transcripts/transcript_traits.h
index 5990fcb95..84eef73aa 100644
--- a/tachyon/crypto/transcripts/transcript_traits.h
+++ b/tachyon/crypto/transcripts/transcript_traits.h
@@ -3,8 +3,8 @@
 
 #include <type_traits>
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
 #include "tachyon/math/finite_fields/prime_field_base.h"
+#include "tachyon/math/geometry/affine_point.h"
 
 namespace tachyon::crypto {
 
diff --git a/tachyon/math/base/semigroups.h b/tachyon/math/base/semigroups.h
index 39c760267..430a90312 100644
--- a/tachyon/math/base/semigroups.h
+++ b/tachyon/math/base/semigroups.h
@@ -72,7 +72,7 @@ struct MultiplicativeSemigroupTraits {
   using ReturnTy = G;
 };
 
-template <typename G>
+template <typename G, typename SFINAE = void>
 struct AdditiveSemigroupTraits {
   using ReturnTy = G;
 };
@@ -320,25 +320,25 @@ class AdditiveSemigroup {
         case 2:
           return Double();
         case 3:
-          return Double() * g;
+          return Double() + g;
         case 4:
           return Double().Double();
         case 5: {
           AddResult g4 = Double();
           g4.DoubleInPlace();
-          return g4 * g;
+          return g4 + g;
         }
         case 6: {
           AddResult g2 = Double();
           AddResult g4 = g2;
           g4.DoubleInPlace();
-          return g4 * g2;
+          return g4 + g2;
         }
         case 7: {
           AddResult g2 = Double();
           AddResult g4 = g2;
           g4.DoubleInPlace();
-          return g4 * g2 * g;
+          return g4 + g2 + g;
         }
         default:
           return DoScalarMul(BigInt<1>(scalar));
diff --git a/tachyon/math/circle/BUILD.bazel b/tachyon/math/circle/BUILD.bazel
index e6f8b08db..68938d7aa 100644
--- a/tachyon/math/circle/BUILD.bazel
+++ b/tachyon/math/circle/BUILD.bazel
@@ -3,26 +3,25 @@ load("//bazel:tachyon_cc.bzl", "tachyon_cc_library", "tachyon_cc_unittest")
 package(default_visibility = ["//visibility:public"])
 
 tachyon_cc_library(
-    name = "circle",
-    hdrs = ["circle.h"],
-    deps = [":circle_traits_forward"],
-)
-
-tachyon_cc_library(
-    name = "circle_point",
-    hdrs = ["circle_point.h"],
+    name = "affine_point",
+    hdrs = ["affine_point.h"],
     deps = [
-        ":circle_point_forward",
         "//tachyon/base:logging",
         "//tachyon/math/base:groups",
+        "//tachyon/math/geometry:curve_type",
         "//tachyon/math/geometry:point2",
         "@com_google_absl//absl/strings",
     ],
 )
 
 tachyon_cc_library(
-    name = "circle_point_forward",
-    hdrs = ["circle_point_forward.h"],
+    name = "circle",
+    hdrs = ["circle.h"],
+    deps = [
+        ":circle_traits_forward",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:curve_type",
+    ],
 )
 
 tachyon_cc_library(
@@ -32,9 +31,9 @@ tachyon_cc_library(
 
 tachyon_cc_unittest(
     name = "circle_unittests",
-    srcs = ["circle_point_unittest.cc"],
+    srcs = ["affine_point_unittest.cc"],
     deps = [
-        "//tachyon/math/circle/stark:g1",
-        "//tachyon/math/circle/stark:g4",
+        "//tachyon/math/circle/m31:g1",
+        "//tachyon/math/circle/m31:g4",
     ],
 )
diff --git a/tachyon/math/circle/circle_point.h b/tachyon/math/circle/affine_point.h
similarity index 50%
rename from tachyon/math/circle/circle_point.h
rename to tachyon/math/circle/affine_point.h
index fb76e34d3..e6e714e41 100644
--- a/tachyon/math/circle/circle_point.h
+++ b/tachyon/math/circle/affine_point.h
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a Apache-2.0 style license that
 // can be found in the LICENSE-APACHE.stwo
 
-#ifndef TACHYON_MATH_CIRCLE_CIRCLE_POINT_H_
-#define TACHYON_MATH_CIRCLE_CIRCLE_POINT_H_
+#ifndef TACHYON_MATH_CIRCLE_AFFINE_POINT_H_
+#define TACHYON_MATH_CIRCLE_AFFINE_POINT_H_
 
 #include <string>
 #include <utility>
@@ -12,61 +12,63 @@
 
 #include "tachyon/base/logging.h"
 #include "tachyon/math/base/groups.h"
-#include "tachyon/math/circle/circle_point_forward.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
 #include "tachyon/math/geometry/point2.h"
 
 namespace tachyon::math {
 
-// TODO(chokobole): Unify this into |AffinePoint|.
-template <typename _Circle>
-class CirclePoint : public AdditiveGroup<CirclePoint<_Circle>> {
+template <typename _Curve>
+class AffinePoint<_Curve, std::enable_if_t<_Curve::kType == CurveType::kCircle>>
+    final : public AdditiveGroup<AffinePoint<_Curve>> {
  public:
-  using Circle = _Circle;
+  using Curve = _Curve;
+  using Circle = _Curve;
   using BaseField = typename Circle::BaseField;
   using ScalarField = typename Circle::ScalarField;
 
-  constexpr CirclePoint() : CirclePoint(BaseField::One(), BaseField::Zero()) {}
-  explicit constexpr CirclePoint(const Point2<BaseField>& point)
-      : CirclePoint(point.x, point.y) {}
-  explicit constexpr CirclePoint(Point2<BaseField>&& point)
-      : CirclePoint(std::move(point.x), std::move(point.y)) {}
-  constexpr CirclePoint(const BaseField& x, const BaseField& y)
+  constexpr AffinePoint() : AffinePoint(BaseField::One(), BaseField::Zero()) {}
+  explicit constexpr AffinePoint(const Point2<BaseField>& point)
+      : AffinePoint(point.x, point.y) {}
+  explicit constexpr AffinePoint(Point2<BaseField>&& point)
+      : AffinePoint(std::move(point.x), std::move(point.y)) {}
+  constexpr AffinePoint(const BaseField& x, const BaseField& y)
       : x_(x), y_(y) {}
-  constexpr CirclePoint(BaseField&& x, BaseField&& y)
+  constexpr AffinePoint(BaseField&& x, BaseField&& y)
       : x_(std::move(x)), y_(std::move(y)) {}
 
-  constexpr static CirclePoint CreateChecked(const BaseField& x,
+  constexpr static AffinePoint CreateChecked(const BaseField& x,
                                              const BaseField& y) {
-    CirclePoint ret = {x, y};
+    AffinePoint ret = {x, y};
     CHECK(ret.IsOnCircle());
     return ret;
   }
 
-  constexpr static CirclePoint CreateChecked(BaseField&& x, BaseField&& y) {
-    CirclePoint ret = {std::move(x), std::move(y)};
+  constexpr static AffinePoint CreateChecked(BaseField&& x, BaseField&& y) {
+    AffinePoint ret = {std::move(x), std::move(y)};
     CHECK(ret.IsOnCircle());
     return ret;
   }
 
-  constexpr static CirclePoint Zero() { return CirclePoint(); }
+  constexpr static AffinePoint Zero() { return AffinePoint(); }
 
-  constexpr static CirclePoint One() { return Generator(); }
+  constexpr static AffinePoint One() { return Generator(); }
 
-  constexpr static CirclePoint Generator() {
+  constexpr static AffinePoint Generator() {
     return {Circle::Config::kGenerator.x, Circle::Config::kGenerator.y};
   }
 
-  constexpr static CirclePoint Random() {
+  constexpr static AffinePoint Random() {
     return ScalarField::Random() * Generator();
   }
 
   constexpr const BaseField& x() const { return x_; }
   constexpr const BaseField& y() const { return y_; }
 
-  constexpr bool operator==(const CirclePoint& other) const {
+  constexpr bool operator==(const AffinePoint& other) const {
     return x_ == other.x_ && y_ == other.y_;
   }
-  constexpr bool operator!=(const CirclePoint& other) const {
+  constexpr bool operator!=(const AffinePoint& other) const {
     return !operator==(other);
   }
 
@@ -74,13 +76,13 @@ class CirclePoint : public AdditiveGroup<CirclePoint<_Circle>> {
 
   constexpr bool IsOnCircle() { return Circle::IsOnCircle(*this); }
 
-  constexpr CirclePoint Conjugate() const { return {x_, -y_}; }
+  constexpr AffinePoint Conjugate() const { return {x_, -y_}; }
 
-  constexpr CirclePoint ComplexConjugate() const {
+  constexpr AffinePoint ComplexConjugate() const {
     return {x_.ComplexConjugate(), y_.ComplexConjugate()};
   }
 
-  constexpr CirclePoint Antipode() const { return {-x_, -y_}; }
+  constexpr AffinePoint Antipode() const { return {-x_, -y_}; }
 
   std::string ToString() const {
     return absl::Substitute("($0, $1)", x_.ToString(), y_.ToString());
@@ -92,33 +94,43 @@ class CirclePoint : public AdditiveGroup<CirclePoint<_Circle>> {
   }
 
   // AdditiveSemigroup methods
-  constexpr CirclePoint Add(const CirclePoint& other) const {
+  constexpr AffinePoint Add(const AffinePoint& other) const {
     return {x_ * other.x_ - y_ * other.y_, x_ * other.y_ + y_ * other.x_};
   }
 
-  constexpr CirclePoint& AddInPlace(const CirclePoint& other) {
+  constexpr AffinePoint& AddInPlace(const AffinePoint& other) {
     return *this = {x_ * other.x_ - y_ * other.y_,
                     x_ * other.y_ + y_ * other.x_};
   }
 
+  constexpr AffinePoint DoubleImpl() const {
+    return {x_.Square().Double() - BaseField::One(), x_.Double() * y_};
+  }
+
+  constexpr AffinePoint& DoubleImplInPlace() {
+    y_ *= x_.Double();
+    x_.SquareInPlace().DoubleInPlace() -= BaseField::One();
+    return *this;
+  }
+
   // AdditiveGroup methods
-  constexpr CirclePoint Sub(const CirclePoint& other) const {
+  constexpr AffinePoint Sub(const AffinePoint& other) const {
     return {x_ * other.x_ + y_ * other.y_, -x_ * other.y_ + y_ * other.x_};
   }
 
-  constexpr CirclePoint& SubInPlace(const CirclePoint& other) {
+  constexpr AffinePoint& SubInPlace(const AffinePoint& other) {
     return *this = {x_ * other.x_ + y_ * other.y_,
                     -x_ * other.y_ + y_ * other.x_};
   }
 
-  constexpr CirclePoint Negate() const { return {x_, -y_}; }
+  constexpr AffinePoint Negate() const { return {x_, -y_}; }
 
-  constexpr CirclePoint& NegateInPlace() {
+  constexpr AffinePoint& NegateInPlace() {
     y_.NegateInPlace();
     return *this;
   }
 
-  constexpr CirclePoint operator*(const ScalarField& idx) const {
+  constexpr AffinePoint operator*(const ScalarField& idx) const {
     return this->ScalarMul(idx.value());
   }
 
@@ -129,4 +141,4 @@ class CirclePoint : public AdditiveGroup<CirclePoint<_Circle>> {
 
 }  // namespace tachyon::math
 
-#endif  // TACHYON_MATH_CIRCLE_CIRCLE_POINT_H_
+#endif  // TACHYON_MATH_CIRCLE_AFFINE_POINT_H_
diff --git a/tachyon/math/circle/affine_point_unittest.cc b/tachyon/math/circle/affine_point_unittest.cc
new file mode 100644
index 000000000..0404a9496
--- /dev/null
+++ b/tachyon/math/circle/affine_point_unittest.cc
@@ -0,0 +1,100 @@
+#include "gtest/gtest.h"
+
+#include "tachyon/math/circle/m31/g1.h"
+#include "tachyon/math/circle/m31/g4.h"
+
+namespace tachyon::math {
+
+template <typename AffinePointType>
+class AffinePointTest : public testing::Test {
+ public:
+  static void SetUpTestSuite() { AffinePointType::Circle::Init(); }
+};
+
+using AffinePointTypes = testing::Types<m31::G1AffinePoint, m31::G4AffinePoint>;
+TYPED_TEST_SUITE(AffinePointTest, AffinePointTypes);
+
+TYPED_TEST(AffinePointTest, IsZero) {
+  using AffinePoint = TypeParam;
+  using BaseField = typename AffinePoint::BaseField;
+
+  EXPECT_TRUE(AffinePoint::Zero().IsZero());
+  EXPECT_FALSE(AffinePoint(BaseField::Zero(), BaseField::One()).IsZero());
+}
+
+TYPED_TEST(AffinePointTest, Generator) {
+  using AffinePoint = TypeParam;
+  using Circle = typename AffinePoint::Circle;
+
+  EXPECT_EQ(
+      AffinePoint::Generator(),
+      AffinePoint(Circle::Config::kGenerator.x, Circle::Config::kGenerator.y));
+}
+
+TYPED_TEST(AffinePointTest, Order) {
+  using AffinePoint = TypeParam;
+
+  AffinePoint r = AffinePoint::Random();
+  EXPECT_TRUE(r.ScalarMul(AffinePoint::ScalarField::Config::kModulus).IsZero());
+}
+
+TYPED_TEST(AffinePointTest, Random) {
+  using AffinePoint = TypeParam;
+
+  bool success = false;
+  AffinePoint r = AffinePoint::Random();
+  for (size_t i = 0; i < 100; ++i) {
+    if (r != AffinePoint::Random()) {
+      success = true;
+      break;
+    }
+  }
+  EXPECT_TRUE(success);
+  EXPECT_TRUE(r.IsOnCircle());
+}
+
+TYPED_TEST(AffinePointTest, EqualityOperators) {
+  using AffinePoint = TypeParam;
+  using BaseField = typename AffinePoint::BaseField;
+
+  AffinePoint p(BaseField::Random(), BaseField::Random());
+  AffinePoint p2(BaseField::Random(), BaseField::Random());
+  EXPECT_EQ(p, p);
+  EXPECT_NE(p, p2);
+}
+
+TYPED_TEST(AffinePointTest, Conjugate) {
+  using AffinePoint = TypeParam;
+
+  AffinePoint p = AffinePoint::Random();
+  EXPECT_EQ(p.Conjugate(), AffinePoint(p.x(), -p.y()));
+}
+
+TYPED_TEST(AffinePointTest, Antipode) {
+  using AffinePoint = TypeParam;
+
+  AffinePoint p = AffinePoint::Random();
+  EXPECT_EQ(p.Antipode(), AffinePoint(-p.x(), -p.y()));
+}
+
+TYPED_TEST(AffinePointTest, AdditiveGroupOperators) {
+  using AffinePoint = TypeParam;
+
+  AffinePoint a = AffinePoint::Random();
+  AffinePoint b = AffinePoint::Random();
+  AffinePoint c = a + b;
+
+  EXPECT_EQ(c - a, b);
+  EXPECT_EQ(c - b, a);
+  AffinePoint doubled = a.Double();
+  EXPECT_EQ(doubled, a + a);
+  AffinePoint a_tmp = a;
+  EXPECT_EQ(a_tmp.DoubleInPlace(), doubled);
+
+  EXPECT_EQ(a + AffinePoint::Zero(), a);
+  a_tmp = a;
+  a_tmp.NegateInPlace();
+  EXPECT_EQ(a_tmp, -a);
+}
+
+}  // namespace tachyon::math
diff --git a/tachyon/math/circle/circle.h b/tachyon/math/circle/circle.h
index 65d166595..0d0190083 100644
--- a/tachyon/math/circle/circle.h
+++ b/tachyon/math/circle/circle.h
@@ -1,8 +1,9 @@
 #ifndef TACHYON_MATH_CIRCLE_CIRCLE_H_
 #define TACHYON_MATH_CIRCLE_CIRCLE_H_
 
-#include "tachyon/math/circle/circle_point.h"
 #include "tachyon/math/circle/circle_traits_forward.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
 
 namespace tachyon::math {
 
@@ -15,7 +16,9 @@ class Circle {
 
   using BaseField = typename Config::BaseField;
   using ScalarField = typename Config::ScalarField;
-  using Point = typename CircleTraits<Config>::PointTy;
+  using AffinePoint = typename CircleTraits<Config>::AffinePointTy;
+
+  constexpr static CurveType kType = CurveType::kCircle;
 
   static void Init() {
     BaseField::Init();
@@ -24,7 +27,7 @@ class Circle {
     Config::Init();
   }
 
-  constexpr static bool IsOnCircle(const Point& point) {
+  constexpr static bool IsOnCircle(const AffinePoint& point) {
     return (point.x().Square() + point.y().Square()).IsOne();
   }
 };
@@ -33,7 +36,7 @@ template <typename Config>
 struct CircleTraits {
   using BaseField = typename Config::BaseField;
   using ScalarField = typename Config::ScalarField;
-  using PointTy = CirclePoint<Circle<Config>>;
+  using AffinePointTy = AffinePoint<Circle<Config>>;
 };
 
 }  // namespace tachyon::math
diff --git a/tachyon/math/circle/circle_point_forward.h b/tachyon/math/circle/circle_point_forward.h
deleted file mode 100644
index 7f654da34..000000000
--- a/tachyon/math/circle/circle_point_forward.h
+++ /dev/null
@@ -1,21 +0,0 @@
-// Copyright 2024 StarkWare Industries Ltd
-// Use of this source code is governed by a Apache-2.0 style license that
-// can be found in the LICENSE-APACHE.stwo
-
-#ifndef TACHYON_MATH_CIRCLE_CIRCLE_POINT_FORWARD_H_
-#define TACHYON_MATH_CIRCLE_CIRCLE_POINT_FORWARD_H_
-
-namespace tachyon::math {
-
-template <typename Circle>
-class CirclePoint;
-
-template <typename ScalarField, typename Circle>
-CirclePoint<Circle> operator*(const ScalarField& v,
-                              const CirclePoint<Circle>& point) {
-  return point * v;
-}
-
-}  // namespace tachyon::math
-
-#endif  // TACHYON_MATH_CIRCLE_CIRCLE_POINT_FORWARD_H_
diff --git a/tachyon/math/circle/circle_point_unittest.cc b/tachyon/math/circle/circle_point_unittest.cc
deleted file mode 100644
index 558689523..000000000
--- a/tachyon/math/circle/circle_point_unittest.cc
+++ /dev/null
@@ -1,98 +0,0 @@
-#include "gtest/gtest.h"
-
-#include "tachyon/math/circle/stark/g1.h"
-#include "tachyon/math/circle/stark/g4.h"
-
-namespace tachyon::math {
-
-template <typename CirclePointType>
-class CirclePointTest : public testing::Test {
- public:
-  static void SetUpTestSuite() { CirclePointType::Circle::Init(); }
-};
-
-using CirclePointTypes =
-    testing::Types<stark::G1CirclePoint, stark::G4CirclePoint>;
-TYPED_TEST_SUITE(CirclePointTest, CirclePointTypes);
-
-TYPED_TEST(CirclePointTest, IsZero) {
-  using CirclePoint = TypeParam;
-  using BaseField = typename CirclePoint::BaseField;
-
-  EXPECT_TRUE(CirclePoint::Zero().IsZero());
-  EXPECT_FALSE(CirclePoint(BaseField::Zero(), BaseField::One()).IsZero());
-}
-
-TYPED_TEST(CirclePointTest, Generator) {
-  using CirclePoint = TypeParam;
-  using Circle = typename CirclePoint::Circle;
-
-  EXPECT_EQ(
-      CirclePoint::Generator(),
-      CirclePoint(Circle::Config::kGenerator.x, Circle::Config::kGenerator.y));
-}
-
-TYPED_TEST(CirclePointTest, Order) {
-  using CirclePoint = TypeParam;
-
-  CirclePoint r = CirclePoint::Random();
-  EXPECT_TRUE(r.ScalarMul(CirclePoint::ScalarField::Config::kModulus).IsZero());
-}
-
-TYPED_TEST(CirclePointTest, Random) {
-  using CirclePoint = TypeParam;
-
-  bool success = false;
-  CirclePoint r = CirclePoint::Random();
-  for (size_t i = 0; i < 100; ++i) {
-    if (r != CirclePoint::Random()) {
-      success = true;
-      break;
-    }
-  }
-  EXPECT_TRUE(success);
-  EXPECT_TRUE(r.IsOnCircle());
-}
-
-TYPED_TEST(CirclePointTest, EqualityOperators) {
-  using CirclePoint = TypeParam;
-  using BaseField = typename CirclePoint::BaseField;
-
-  CirclePoint p(BaseField::Random(), BaseField::Random());
-  CirclePoint p2(BaseField::Random(), BaseField::Random());
-  EXPECT_EQ(p, p);
-  EXPECT_NE(p, p2);
-}
-
-TYPED_TEST(CirclePointTest, Conjugate) {
-  using CirclePoint = TypeParam;
-
-  CirclePoint p = CirclePoint::Random();
-  EXPECT_EQ(p.Conjugate(), CirclePoint(p.x(), -p.y()));
-}
-
-TYPED_TEST(CirclePointTest, Antipode) {
-  using CirclePoint = TypeParam;
-
-  CirclePoint p = CirclePoint::Random();
-  EXPECT_EQ(p.Antipode(), CirclePoint(-p.x(), -p.y()));
-}
-
-TYPED_TEST(CirclePointTest, AdditiveGroupOperators) {
-  using CirclePoint = TypeParam;
-
-  CirclePoint a = CirclePoint::Random();
-  CirclePoint b = CirclePoint::Random();
-  CirclePoint c = a + b;
-
-  EXPECT_EQ(c - a, b);
-  EXPECT_EQ(c - b, a);
-  EXPECT_EQ(a.Double(), a + a);
-
-  EXPECT_EQ(a + CirclePoint::Zero(), a);
-  CirclePoint a_tmp = a;
-  a_tmp.NegateInPlace();
-  EXPECT_EQ(a_tmp, -a);
-}
-
-}  // namespace tachyon::math
diff --git a/tachyon/math/circle/generator/build_defs.bzl b/tachyon/math/circle/generator/build_defs.bzl
index 15dc967a2..d7dd75af3 100644
--- a/tachyon/math/circle/generator/build_defs.bzl
+++ b/tachyon/math/circle/generator/build_defs.bzl
@@ -96,8 +96,8 @@ def generate_circle_points(
         deps = [
             base_field_dep,
             scalar_field_dep,
+            "//tachyon/math/circle:affine_point",
             "//tachyon/math/circle:circle",
-            "//tachyon/math/circle:circle_point",
         ],
         **kwargs
     )
diff --git a/tachyon/math/circle/generator/cpu.h.tpl b/tachyon/math/circle/generator/cpu.h.tpl
index 38aa66bbc..32d9f7ab8 100644
--- a/tachyon/math/circle/generator/cpu.h.tpl
+++ b/tachyon/math/circle/generator/cpu.h.tpl
@@ -1,7 +1,7 @@
 // clang-format off
 #include "tachyon/base/logging.h"
+#include "tachyon/math/circle/affine_point.h"
 #include "tachyon/math/circle/circle.h"
-#include "tachyon/math/circle/circle_point.h"
 #include "tachyon/math/geometry/point2.h"
 #include "%{base_field_hdr}"
 #include "%{scalar_field_hdr}"
@@ -29,7 +29,7 @@ template <typename BaseField, typename ScalarField>
 Point2<BaseField> %{class}CircleConfig<BaseField, ScalarField>::kGenerator;
 
 using %{class}Circle = tachyon::math::Circle<%{class}CircleConfig<%{base_field}, %{scalar_field}>>;
-using %{class}CirclePoint = tachyon::math::CirclePoint<%{class}Circle>;
+using %{class}AffinePoint = tachyon::math::AffinePoint<%{class}Circle>;
 
 }  // namespace %{namespace}
 // clang-format on
diff --git a/tachyon/math/circle/stark/BUILD.bazel b/tachyon/math/circle/m31/BUILD.bazel
similarity index 81%
rename from tachyon/math/circle/stark/BUILD.bazel
rename to tachyon/math/circle/m31/BUILD.bazel
index 49bdc62e5..8c4dd009f 100644
--- a/tachyon/math/circle/stark/BUILD.bazel
+++ b/tachyon/math/circle/m31/BUILD.bazel
@@ -15,10 +15,10 @@ generate_circle_points(
     base_field_dep = "//tachyon/math/finite_fields/mersenne31",
     base_field_hdr = "tachyon/math/finite_fields/mersenne31/mersenne31.h",
     class_name = "G1",
-    namespace = "tachyon::math::stark",
+    namespace = "tachyon::math::m31",
     scalar_field = "Fr",
     scalar_field_dep = ":fr",
-    scalar_field_hdr = "tachyon/math/circle/stark/fr.h",
+    scalar_field_hdr = "tachyon/math/circle/m31/fr.h",
     x = ["2"],
     y = ["1268011823"],
 )
@@ -29,7 +29,7 @@ generate_prime_fields(
     # 2³¹
     # Hex: 0x80000000
     modulus = "2147483648",
-    namespace = "tachyon::math::stark",
+    namespace = "tachyon::math::m31",
     reduce32 = "return v & (kModulus - 1);",
     reduce64 = "return v & (kModulus - 1);",
     use_montgomery = False,
@@ -41,7 +41,7 @@ generate_fp2s(
     base_field_hdr = "tachyon/math/finite_fields/mersenne31/mersenne31.h",
     class_name = "Fq2",
     is_packed = False,
-    namespace = "tachyon::math::stark",
+    namespace = "tachyon::math::m31",
     non_residue = ["-1"],
     deps = ["//tachyon/math/finite_fields/mersenne31"],
 )
@@ -50,10 +50,10 @@ generate_fp4s(
     name = "fq4",
     base_field = "Fq2",
     base_field_degree = 2,
-    base_field_hdr = "tachyon/math/circle/stark/fq2.h",
+    base_field_hdr = "tachyon/math/circle/m31/fq2.h",
     class_name = "Fq4",
     is_packed = False,
-    namespace = "tachyon::math::stark",
+    namespace = "tachyon::math::m31",
     non_residue = [
         "2",
         "1",
@@ -67,7 +67,7 @@ generate_prime_fields(
     # (2³¹ - 1)⁴ - 1
     # Hex: 0xfffffff800000017ffffffe00000000
     modulus = "21267647892944572736998860269687930880",
-    namespace = "tachyon::math::stark",
+    namespace = "tachyon::math::m31",
     use_asm = False,
 )
 
@@ -76,12 +76,12 @@ generate_circle_points(
     base_field = "Fq4",
     base_field_degree = 4,
     base_field_dep = ":fq4",
-    base_field_hdr = "tachyon/math/circle/stark/fq4.h",
+    base_field_hdr = "tachyon/math/circle/m31/fq4.h",
     class_name = "G4",
-    namespace = "tachyon::math::stark",
+    namespace = "tachyon::math::m31",
     scalar_field = "Fr4",
     scalar_field_dep = ":fr4",
-    scalar_field_hdr = "tachyon/math/circle/stark/fr4.h",
+    scalar_field_hdr = "tachyon/math/circle/m31/fr4.h",
     x = [
         "1",
         "0",
diff --git a/tachyon/math/elliptic_curves/BUILD.bazel b/tachyon/math/elliptic_curves/BUILD.bazel
index 784590acd..a37960be0 100644
--- a/tachyon/math/elliptic_curves/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/BUILD.bazel
@@ -3,22 +3,12 @@ load("//bazel:tachyon_cc.bzl", "tachyon_cc_library")
 package(default_visibility = ["//visibility:public"])
 
 tachyon_cc_library(
-    name = "points",
-    hdrs = [
-        "affine_point.h",
-        "curve_type.h",
-        "jacobian_point.h",
-        "point_conversions.h",
-        "point_conversions_forward.h",
-        "point_xyzz.h",
-        "projective_point.h",
-        "semigroups.h",
-    ],
+    name = "semigroups",
+    hdrs = ["semigroups.h"],
     deps = [
-        "//tachyon/base:template_util",
-        "//tachyon/base/containers:container_util",
-        "//tachyon/math/geometry:point2",
-        "//tachyon/math/geometry:point3",
-        "//tachyon/math/geometry:point4",
+        "//tachyon/math/base:semigroups",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:curve_type",
+        "//tachyon/math/geometry:jacobian_point",
     ],
 )
diff --git a/tachyon/math/elliptic_curves/curve_type.h b/tachyon/math/elliptic_curves/curve_type.h
deleted file mode 100644
index ef719e058..000000000
--- a/tachyon/math/elliptic_curves/curve_type.h
+++ /dev/null
@@ -1,13 +0,0 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_CURVE_TYPE_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_CURVE_TYPE_H_
-
-namespace tachyon::math {
-
-enum class CurveType {
-  kShortWeierstrass,
-  kTwistedEdwards,
-};
-
-}  // namespace tachyon::math
-
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_CURVE_TYPE_H_
diff --git a/tachyon/math/elliptic_curves/msm/BUILD.bazel b/tachyon/math/elliptic_curves/msm/BUILD.bazel
index 218779a40..05ce60208 100644
--- a/tachyon/math/elliptic_curves/msm/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/msm/BUILD.bazel
@@ -12,7 +12,7 @@ tachyon_cc_library(
         "//tachyon/base/containers:container_util",
         "//tachyon/math/base:bit_iterator",
         "//tachyon/math/base:semigroups",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:point_conversions",
     ],
 )
 
@@ -23,7 +23,7 @@ tachyon_cc_library(
         "//tachyon/math/base:bit_iterator",
         "//tachyon/math/base/gmp:bit_traits",
         "//tachyon/math/base/gmp:signed_value",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/elliptic_curves:semigroups",
         "//tachyon/math/matrix:gmp_num_traits",
     ],
 )
diff --git a/tachyon/math/elliptic_curves/msm/algorithms/icicle/BUILD.bazel b/tachyon/math/elliptic_curves/msm/algorithms/icicle/BUILD.bazel
index 0a784c362..727818232 100644
--- a/tachyon/math/elliptic_curves/msm/algorithms/icicle/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/msm/algorithms/icicle/BUILD.bazel
@@ -31,8 +31,8 @@ tachyon_cc_library(
         "//tachyon/base:bit_cast",
         "//tachyon/device/gpu:gpu_device_functions",
         "//tachyon/device/gpu:gpu_enums",
-        "//tachyon/math/elliptic_curves:points",
         "//tachyon/math/elliptic_curves/bn/bn254:g1",
         "//tachyon/math/elliptic_curves/bn/bn254:g2",
+        "//tachyon/math/geometry:jacobian_point",
     ],
 )
diff --git a/tachyon/math/elliptic_curves/msm/algorithms/icicle/icicle_msm.h b/tachyon/math/elliptic_curves/msm/algorithms/icicle/icicle_msm.h
index 1969d7a01..4ae6de232 100644
--- a/tachyon/math/elliptic_curves/msm/algorithms/icicle/icicle_msm.h
+++ b/tachyon/math/elliptic_curves/msm/algorithms/icicle/icicle_msm.h
@@ -13,7 +13,7 @@
 #include "tachyon/math/elliptic_curves/bn/bn254/g2.h"
 #include "tachyon/math/elliptic_curves/msm/algorithms/icicle/icicle_msm_bn254_g1.h"
 #include "tachyon/math/elliptic_curves/msm/algorithms/icicle/icicle_msm_bn254_g2.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/msm/algorithms/pippenger/BUILD.bazel b/tachyon/math/elliptic_curves/msm/algorithms/pippenger/BUILD.bazel
index 71bee0164..581ffe4bd 100644
--- a/tachyon/math/elliptic_curves/msm/algorithms/pippenger/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/msm/algorithms/pippenger/BUILD.bazel
@@ -13,6 +13,7 @@ tachyon_cc_library(
     deps = [
         ":pippenger_base",
         "//tachyon/base:openmp_util",
+        "//tachyon/math/elliptic_curves:semigroups",
         "//tachyon/math/elliptic_curves/msm:msm_ctx",
         "//tachyon/math/elliptic_curves/msm:msm_util",
     ],
@@ -30,7 +31,8 @@ tachyon_cc_library(
     deps = [
         "//tachyon/base/containers:adapters",
         "//tachyon/math/base:semigroups",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:point_xyzz",
         "@com_google_absl//absl/types:span",
     ],
 )
diff --git a/tachyon/math/elliptic_curves/msm/algorithms/pippenger/pippenger_base.h b/tachyon/math/elliptic_curves/msm/algorithms/pippenger/pippenger_base.h
index 2c3f034a6..8d4b2d1f3 100644
--- a/tachyon/math/elliptic_curves/msm/algorithms/pippenger/pippenger_base.h
+++ b/tachyon/math/elliptic_curves/msm/algorithms/pippenger/pippenger_base.h
@@ -10,8 +10,8 @@
 
 #include "tachyon/base/containers/adapters.h"
 #include "tachyon/math/base/semigroups.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/point_xyzz.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/msm/fixed_base_msm.h b/tachyon/math/elliptic_curves/msm/fixed_base_msm.h
index ce3110759..2c09ec096 100644
--- a/tachyon/math/elliptic_curves/msm/fixed_base_msm.h
+++ b/tachyon/math/elliptic_curves/msm/fixed_base_msm.h
@@ -15,7 +15,7 @@
 #include "tachyon/math/base/bit_iterator.h"
 #include "tachyon/math/base/semigroups.h"
 #include "tachyon/math/elliptic_curves/msm/msm_ctx.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/msm/glv.h b/tachyon/math/elliptic_curves/msm/glv.h
index efa3a78e9..a9925d20d 100644
--- a/tachyon/math/elliptic_curves/msm/glv.h
+++ b/tachyon/math/elliptic_curves/msm/glv.h
@@ -9,10 +9,6 @@
 #include "tachyon/math/base/bit_iterator.h"
 #include "tachyon/math/base/gmp/bit_traits.h"
 #include "tachyon/math/base/gmp/signed_value.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
 #include "tachyon/math/elliptic_curves/semigroups.h"
 #include "tachyon/math/matrix/gmp_num_traits.h"
 
diff --git a/tachyon/math/elliptic_curves/msm/glv_unittest.cc b/tachyon/math/elliptic_curves/msm/glv_unittest.cc
index 8800adc13..4c6a240ea 100644
--- a/tachyon/math/elliptic_curves/msm/glv_unittest.cc
+++ b/tachyon/math/elliptic_curves/msm/glv_unittest.cc
@@ -6,7 +6,7 @@
 #include "tachyon/math/elliptic_curves/bls12/bls12_381/g2.h"
 #include "tachyon/math/elliptic_curves/bn/bn254/g1.h"
 #include "tachyon/math/elliptic_curves/bn/bn254/g2.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/msm/test/fixed_base_msm_test_set.h b/tachyon/math/elliptic_curves/msm/test/fixed_base_msm_test_set.h
index f29b5675a..00d4f5195 100644
--- a/tachyon/math/elliptic_curves/msm/test/fixed_base_msm_test_set.h
+++ b/tachyon/math/elliptic_curves/msm/test/fixed_base_msm_test_set.h
@@ -8,8 +8,8 @@
 #include "tachyon/base/logging.h"
 #include "tachyon/math/base/semigroups.h"
 #include "tachyon/math/elliptic_curves/msm/fixed_base_msm.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
 #include "tachyon/math/elliptic_curves/test/random.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/msm/test/variable_base_msm_test_set.h b/tachyon/math/elliptic_curves/msm/test/variable_base_msm_test_set.h
index 56a0fe6d0..1483b09f5 100644
--- a/tachyon/math/elliptic_curves/msm/test/variable_base_msm_test_set.h
+++ b/tachyon/math/elliptic_curves/msm/test/variable_base_msm_test_set.h
@@ -8,8 +8,8 @@
 #include "tachyon/base/logging.h"
 #include "tachyon/math/base/semigroups.h"
 #include "tachyon/math/elliptic_curves/msm/variable_base_msm.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
 #include "tachyon/math/elliptic_curves/test/random.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/point_conversions_forward.h b/tachyon/math/elliptic_curves/point_conversions_forward.h
deleted file mode 100644
index 9b1a1c462..000000000
--- a/tachyon/math/elliptic_curves/point_conversions_forward.h
+++ /dev/null
@@ -1,11 +0,0 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_POINT_CONVERSIONS_FORWARD_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_POINT_CONVERSIONS_FORWARD_H_
-
-namespace tachyon::math {
-
-template <typename SrcPoint, typename DstPoint, typename SFINAE = void>
-struct PointConversions;
-
-}  // namespace tachyon::math
-
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_POINT_CONVERSIONS_FORWARD_H_
diff --git a/tachyon/math/elliptic_curves/semigroups.h b/tachyon/math/elliptic_curves/semigroups.h
index ea8fe7aa4..3a3196443 100644
--- a/tachyon/math/elliptic_curves/semigroups.h
+++ b/tachyon/math/elliptic_curves/semigroups.h
@@ -2,13 +2,16 @@
 #define TACHYON_MATH_ELLIPTIC_CURVES_SEMIGROUPS_H_
 
 #include "tachyon/math/base/semigroups.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
+#include "tachyon/math/geometry/jacobian_point.h"
 
 namespace tachyon::math::internal {
 
 template <typename Curve>
-struct AdditiveSemigroupTraits<AffinePoint<Curve>> {
+struct AdditiveSemigroupTraits<
+    AffinePoint<Curve>,
+    std::enable_if_t<Curve::kType == CurveType::kShortWeierstrass>> {
   using ReturnTy = JacobianPoint<Curve>;
 };
 
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/BUILD.bazel b/tachyon/math/elliptic_curves/short_weierstrass/BUILD.bazel
index 6da4d6cd4..7320a8931 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/short_weierstrass/BUILD.bazel
@@ -26,10 +26,15 @@ tachyon_cc_library(
         "//tachyon/base/containers:container_util",
         "//tachyon/base/json",
         "//tachyon/math/base:groups",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/elliptic_curves:semigroups",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:curve_type",
+        "//tachyon/math/geometry:jacobian_point",
         "//tachyon/math/geometry:point2",
         "//tachyon/math/geometry:point3",
         "//tachyon/math/geometry:point4",
+        "//tachyon/math/geometry:point_xyzz",
+        "//tachyon/math/geometry:projective_point",
     ],
 )
 
@@ -38,7 +43,11 @@ tachyon_cc_library(
     srcs = ["sw_curve.h"],
     deps = [
         ":sw_curve_traits_forward",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:curve_type",
+        "//tachyon/math/geometry:jacobian_point",
+        "//tachyon/math/geometry:point_conversions",
+        "//tachyon/math/geometry:projective_point",
     ],
 )
 
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/affine_point.h b/tachyon/math/elliptic_curves/short_weierstrass/affine_point.h
index be2fe803c..4a9eec0c4 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/affine_point.h
+++ b/tachyon/math/elliptic_curves/short_weierstrass/affine_point.h
@@ -14,13 +14,13 @@
 #include "tachyon/base/logging.h"
 #include "tachyon/base/parallelize.h"
 #include "tachyon/math/base/groups.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/curve_type.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
 #include "tachyon/math/elliptic_curves/semigroups.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
+#include "tachyon/math/geometry/jacobian_point.h"
 #include "tachyon/math/geometry/point2.h"
+#include "tachyon/math/geometry/point_xyzz.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon {
 namespace math {
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/affine_point_correctness_gpu_test.cc b/tachyon/math/elliptic_curves/short_weierstrass/affine_point_correctness_gpu_test.cc
index 3880f84df..3475e54d6 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/affine_point_correctness_gpu_test.cc
+++ b/tachyon/math/elliptic_curves/short_weierstrass/affine_point_correctness_gpu_test.cc
@@ -3,8 +3,8 @@
 
 #include "tachyon/device/gpu/gpu_memory.h"
 #include "tachyon/math/elliptic_curves/bn/bn254/g1_gpu.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
 #include "tachyon/math/elliptic_curves/short_weierstrass/kernels/elliptic_curve_ops.cu.h"
+#include "tachyon/math/geometry/point_conversions.h"
 #include "tachyon/math/test/launch_op_macros.h"
 
 namespace tachyon::math {
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/jacobian_point.h b/tachyon/math/elliptic_curves/short_weierstrass/jacobian_point.h
index fc6894f0d..5e1099fff 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/jacobian_point.h
+++ b/tachyon/math/elliptic_curves/short_weierstrass/jacobian_point.h
@@ -14,12 +14,12 @@
 #include "tachyon/base/json/json.h"
 #include "tachyon/base/logging.h"
 #include "tachyon/math/base/groups.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/curve_type.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
+#include "tachyon/math/geometry/jacobian_point.h"
 #include "tachyon/math/geometry/point3.h"
+#include "tachyon/math/geometry/point_xyzz.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon {
 namespace math {
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/non_affine_point_correctness_gpu_test.cc b/tachyon/math/elliptic_curves/short_weierstrass/non_affine_point_correctness_gpu_test.cc
index de374b527..53d162d98 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/non_affine_point_correctness_gpu_test.cc
+++ b/tachyon/math/elliptic_curves/short_weierstrass/non_affine_point_correctness_gpu_test.cc
@@ -3,8 +3,8 @@
 
 #include "tachyon/device/gpu/gpu_memory.h"
 #include "tachyon/math/elliptic_curves/bn/bn254/g1_gpu.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
 #include "tachyon/math/elliptic_curves/short_weierstrass/kernels/elliptic_curve_ops.cu.h"
+#include "tachyon/math/geometry/point_conversions.h"
 #include "tachyon/math/test/launch_op_macros.h"
 
 namespace tachyon::math {
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/point_xyzz.h b/tachyon/math/elliptic_curves/short_weierstrass/point_xyzz.h
index 67bc583a3..0b9de58f4 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/point_xyzz.h
+++ b/tachyon/math/elliptic_curves/short_weierstrass/point_xyzz.h
@@ -13,11 +13,12 @@
 #include "tachyon/base/json/json.h"
 #include "tachyon/base/logging.h"
 #include "tachyon/math/base/groups.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/curve_type.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
+#include "tachyon/math/geometry/jacobian_point.h"
 #include "tachyon/math/geometry/point4.h"
+#include "tachyon/math/geometry/point_xyzz.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon {
 namespace math {
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/projective_point.h b/tachyon/math/elliptic_curves/short_weierstrass/projective_point.h
index 7bf5f5acb..dba9e37ea 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/projective_point.h
+++ b/tachyon/math/elliptic_curves/short_weierstrass/projective_point.h
@@ -11,12 +11,12 @@
 #include "tachyon/base/containers/container_util.h"
 #include "tachyon/base/logging.h"
 #include "tachyon/math/base/groups.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/curve_type.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
+#include "tachyon/math/geometry/jacobian_point.h"
 #include "tachyon/math/geometry/point3.h"
+#include "tachyon/math/geometry/point_xyzz.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon {
 namespace math {
diff --git a/tachyon/math/elliptic_curves/short_weierstrass/sw_curve.h b/tachyon/math/elliptic_curves/short_weierstrass/sw_curve.h
index cb60d9358..f413d42a0 100644
--- a/tachyon/math/elliptic_curves/short_weierstrass/sw_curve.h
+++ b/tachyon/math/elliptic_curves/short_weierstrass/sw_curve.h
@@ -3,13 +3,13 @@
 
 #include <utility>
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/curve_type.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
 #include "tachyon/math/elliptic_curves/short_weierstrass/sw_curve_traits_forward.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/curve_type.h"
+#include "tachyon/math/geometry/jacobian_point.h"
+#include "tachyon/math/geometry/point_conversions.h"
+#include "tachyon/math/geometry/point_xyzz.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/elliptic_curves/test/BUILD.bazel b/tachyon/math/elliptic_curves/test/BUILD.bazel
index 424a7ddee..4f2f76faf 100644
--- a/tachyon/math/elliptic_curves/test/BUILD.bazel
+++ b/tachyon/math/elliptic_curves/test/BUILD.bazel
@@ -8,6 +8,6 @@ tachyon_cc_library(
     hdrs = ["random.h"],
     deps = [
         "//tachyon/base:parallelize",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:point_conversions",
     ],
 )
diff --git a/tachyon/math/elliptic_curves/test/random.h b/tachyon/math/elliptic_curves/test/random.h
index 057b3d65d..82c7bcf01 100644
--- a/tachyon/math/elliptic_curves/test/random.h
+++ b/tachyon/math/elliptic_curves/test/random.h
@@ -4,7 +4,7 @@
 #include <vector>
 
 #include "tachyon/base/parallelize.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/point_conversions.h"
 
 namespace tachyon::math {
 
diff --git a/tachyon/math/geometry/BUILD.bazel b/tachyon/math/geometry/BUILD.bazel
index 923a08551..4c2342c9c 100644
--- a/tachyon/math/geometry/BUILD.bazel
+++ b/tachyon/math/geometry/BUILD.bazel
@@ -2,6 +2,17 @@ load("//bazel:tachyon_cc.bzl", "tachyon_cc_library", "tachyon_cc_unittest")
 
 package(default_visibility = ["//visibility:public"])
 
+tachyon_cc_library(
+    name = "affine_point",
+    hdrs = ["affine_point.h"],
+    deps = [":point_conversions_forward"],
+)
+
+tachyon_cc_library(
+    name = "curve_type",
+    hdrs = ["curve_type.h"],
+)
+
 tachyon_cc_library(
     name = "dimensions",
     hdrs = ["dimensions.h"],
@@ -13,6 +24,15 @@ tachyon_cc_library(
     ],
 )
 
+tachyon_cc_library(
+    name = "jacobian_point",
+    hdrs = ["jacobian_point.h"],
+    deps = [
+        ":point3",
+        ":point_conversions_forward",
+    ],
+)
+
 tachyon_cc_library(
     name = "point2",
     hdrs = ["point2.h"],
@@ -43,6 +63,42 @@ tachyon_cc_library(
     ],
 )
 
+tachyon_cc_library(
+    name = "point_conversions_forward",
+    hdrs = ["point_conversions_forward.h"],
+)
+
+tachyon_cc_library(
+    name = "point_conversions",
+    hdrs = ["point_conversions.h"],
+    deps = [
+        ":affine_point",
+        ":jacobian_point",
+        ":point_conversions_forward",
+        ":point_xyzz",
+        ":projective_point",
+        "//tachyon/base:template_util",
+    ],
+)
+
+tachyon_cc_library(
+    name = "point_xyzz",
+    hdrs = ["point_xyzz.h"],
+    deps = [
+        ":point4",
+        ":point_conversions_forward",
+    ],
+)
+
+tachyon_cc_library(
+    name = "projective_point",
+    hdrs = ["projective_point.h"],
+    deps = [
+        ":point3",
+        ":point_conversions_forward",
+    ],
+)
+
 tachyon_cc_unittest(
     name = "geometry_unittests",
     srcs = [
diff --git a/tachyon/math/elliptic_curves/affine_point.h b/tachyon/math/geometry/affine_point.h
similarity index 70%
rename from tachyon/math/elliptic_curves/affine_point.h
rename to tachyon/math/geometry/affine_point.h
index e215bf360..c65788aeb 100644
--- a/tachyon/math/elliptic_curves/affine_point.h
+++ b/tachyon/math/geometry/affine_point.h
@@ -1,10 +1,9 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_AFFINE_POINT_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_AFFINE_POINT_H_
+#ifndef TACHYON_MATH_GEOMETRY_AFFINE_POINT_H_
+#define TACHYON_MATH_GEOMETRY_AFFINE_POINT_H_
 
 #include <utility>
 
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_conversions_forward.h"
+#include "tachyon/math/geometry/point_conversions_forward.h"
 
 namespace tachyon::math {
 
@@ -14,8 +13,7 @@ class AffinePoint;
 template <typename ScalarField, typename Curve,
           std::enable_if_t<std::is_same_v<
               ScalarField, typename Curve::ScalarField>>* = nullptr>
-JacobianPoint<Curve> operator*(const ScalarField& v,
-                               const AffinePoint<Curve>& point) {
+auto operator*(const ScalarField& v, const AffinePoint<Curve>& point) {
   return point * v;
 }
 
@@ -39,4 +37,4 @@ struct PointConversions<AffinePoint<SrcCurve>, AffinePoint<DstCurve>,
 
 }  // namespace tachyon::math
 
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_AFFINE_POINT_H_
+#endif  // TACHYON_MATH_GEOMETRY_AFFINE_POINT_H_
diff --git a/tachyon/math/geometry/curve_type.h b/tachyon/math/geometry/curve_type.h
new file mode 100644
index 000000000..fe07e026f
--- /dev/null
+++ b/tachyon/math/geometry/curve_type.h
@@ -0,0 +1,14 @@
+#ifndef TACHYON_MATH_GEOMETRY_CURVE_TYPE_H_
+#define TACHYON_MATH_GEOMETRY_CURVE_TYPE_H_
+
+namespace tachyon::math {
+
+enum class CurveType {
+  kCircle,
+  kShortWeierstrass,
+  kTwistedEdwards,
+};
+
+}  // namespace tachyon::math
+
+#endif  // TACHYON_MATH_GEOMETRY_CURVE_TYPE_H_
diff --git a/tachyon/math/elliptic_curves/jacobian_point.h b/tachyon/math/geometry/jacobian_point.h
similarity index 83%
rename from tachyon/math/elliptic_curves/jacobian_point.h
rename to tachyon/math/geometry/jacobian_point.h
index fd9dc0ff4..917fcc0b3 100644
--- a/tachyon/math/elliptic_curves/jacobian_point.h
+++ b/tachyon/math/geometry/jacobian_point.h
@@ -1,10 +1,10 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_JACOBIAN_POINT_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_JACOBIAN_POINT_H_
+#ifndef TACHYON_MATH_GEOMETRY_JACOBIAN_POINT_H_
+#define TACHYON_MATH_GEOMETRY_JACOBIAN_POINT_H_
 
 #include <utility>
 
-#include "tachyon/math/elliptic_curves/point_conversions_forward.h"
 #include "tachyon/math/geometry/point3.h"
+#include "tachyon/math/geometry/point_conversions_forward.h"
 
 namespace tachyon::math {
 
@@ -39,4 +39,4 @@ struct PointConversions<JacobianPoint<SrcCurve>, JacobianPoint<DstCurve>,
 
 }  // namespace tachyon::math
 
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_JACOBIAN_POINT_H_
+#endif  // TACHYON_MATH_GEOMETRY_JACOBIAN_POINT_H_
diff --git a/tachyon/math/elliptic_curves/point_conversions.h b/tachyon/math/geometry/point_conversions.h
similarity index 87%
rename from tachyon/math/elliptic_curves/point_conversions.h
rename to tachyon/math/geometry/point_conversions.h
index bbad02f32..9e2edf8ed 100644
--- a/tachyon/math/elliptic_curves/point_conversions.h
+++ b/tachyon/math/geometry/point_conversions.h
@@ -1,13 +1,12 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_POINT_CONVERSIONS_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_POINT_CONVERSIONS_H_
+#ifndef TACHYON_MATH_GEOMETRY_POINT_CONVERSIONS_H_
+#define TACHYON_MATH_GEOMETRY_POINT_CONVERSIONS_H_
 
-#include "tachyon/base/containers/container_util.h"
 #include "tachyon/base/template_util.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
-#include "tachyon/math/elliptic_curves/jacobian_point.h"
-#include "tachyon/math/elliptic_curves/point_conversions_forward.h"
-#include "tachyon/math/elliptic_curves/point_xyzz.h"
-#include "tachyon/math/elliptic_curves/projective_point.h"
+#include "tachyon/math/geometry/affine_point.h"
+#include "tachyon/math/geometry/jacobian_point.h"
+#include "tachyon/math/geometry/point_conversions_forward.h"
+#include "tachyon/math/geometry/point_xyzz.h"
+#include "tachyon/math/geometry/projective_point.h"
 
 namespace tachyon::math {
 
@@ -127,4 +126,4 @@ struct PointConversions<PointXYZZ<Curve>, JacobianPoint<Curve>> {
 
 }  // namespace tachyon::math
 
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_POINT_CONVERSIONS_H_
+#endif  // TACHYON_MATH_GEOMETRY_POINT_CONVERSIONS_H_
diff --git a/tachyon/math/geometry/point_conversions_forward.h b/tachyon/math/geometry/point_conversions_forward.h
new file mode 100644
index 000000000..689602e18
--- /dev/null
+++ b/tachyon/math/geometry/point_conversions_forward.h
@@ -0,0 +1,11 @@
+#ifndef TACHYON_MATH_GEOMETRY_POINT_CONVERSIONS_FORWARD_H_
+#define TACHYON_MATH_GEOMETRY_POINT_CONVERSIONS_FORWARD_H_
+
+namespace tachyon::math {
+
+template <typename SrcPoint, typename DstPoint, typename SFINAE = void>
+struct PointConversions;
+
+}  // namespace tachyon::math
+
+#endif  // TACHYON_MATH_GEOMETRY_POINT_CONVERSIONS_FORWARD_H_
diff --git a/tachyon/math/elliptic_curves/point_xyzz.h b/tachyon/math/geometry/point_xyzz.h
similarity index 83%
rename from tachyon/math/elliptic_curves/point_xyzz.h
rename to tachyon/math/geometry/point_xyzz.h
index 11c7593c4..90c7e38e1 100644
--- a/tachyon/math/elliptic_curves/point_xyzz.h
+++ b/tachyon/math/geometry/point_xyzz.h
@@ -1,10 +1,10 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_POINT_XYZZ_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_POINT_XYZZ_H_
+#ifndef TACHYON_MATH_GEOMETRY_POINT_XYZZ_H_
+#define TACHYON_MATH_GEOMETRY_POINT_XYZZ_H_
 
 #include <utility>
 
-#include "tachyon/math/elliptic_curves/point_conversions_forward.h"
 #include "tachyon/math/geometry/point4.h"
+#include "tachyon/math/geometry/point_conversions_forward.h"
 
 namespace tachyon::math {
 
@@ -39,4 +39,4 @@ struct PointConversions<PointXYZZ<SrcCurve>, PointXYZZ<DstCurve>,
 
 }  // namespace tachyon::math
 
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_POINT_XYZZ_H_
+#endif  // TACHYON_MATH_GEOMETRY_POINT_XYZZ_H_
diff --git a/tachyon/math/elliptic_curves/projective_point.h b/tachyon/math/geometry/projective_point.h
similarity index 80%
rename from tachyon/math/elliptic_curves/projective_point.h
rename to tachyon/math/geometry/projective_point.h
index 5a72724c5..81ffb8aa6 100644
--- a/tachyon/math/elliptic_curves/projective_point.h
+++ b/tachyon/math/geometry/projective_point.h
@@ -1,11 +1,10 @@
-#ifndef TACHYON_MATH_ELLIPTIC_CURVES_PROJECTIVE_POINT_H_
-#define TACHYON_MATH_ELLIPTIC_CURVES_PROJECTIVE_POINT_H_
+#ifndef TACHYON_MATH_GEOMETRY_PROJECTIVE_POINT_H_
+#define TACHYON_MATH_GEOMETRY_PROJECTIVE_POINT_H_
 
 #include <utility>
 
-#include "tachyon/base/buffer/copyable.h"
-#include "tachyon/math/elliptic_curves/point_conversions_forward.h"
 #include "tachyon/math/geometry/point3.h"
+#include "tachyon/math/geometry/point_conversions_forward.h"
 
 namespace tachyon::math {
 
@@ -40,4 +39,4 @@ struct PointConversions<ProjectivePoint<SrcCurve>, ProjectivePoint<DstCurve>,
 
 }  // namespace tachyon::math
 
-#endif  // TACHYON_MATH_ELLIPTIC_CURVES_PROJECTIVE_POINT_H_
+#endif  // TACHYON_MATH_GEOMETRY_PROJECTIVE_POINT_H_
diff --git a/tachyon/math/polynomials/univariate/univariate_dense_coefficients.h b/tachyon/math/polynomials/univariate/univariate_dense_coefficients.h
index 05e6f8298..812e4732c 100644
--- a/tachyon/math/polynomials/univariate/univariate_dense_coefficients.h
+++ b/tachyon/math/polynomials/univariate/univariate_dense_coefficients.h
@@ -173,27 +173,18 @@ class UnivariateDenseCoefficients {
 
   // Return coefficients where the original coefficients reduce their degree
   // by categorizing coefficients into even and odd degrees,
-  // multiplying either set of coefficients by a specified random field |r|,
+  // multiplying coefficients with odd degrees by a specified random field |r|,
   // and summing them together.
-  template <bool MulRandomWithEvens>
   CONSTEXPR_IF_NOT_OPENMP UnivariateDenseCoefficients
   Fold(const Field& r) const {
     size_t size = coefficients_.size();
     std::vector<F> coefficients((size + 1) >> 1);
     OPENMP_PARALLEL_FOR(size_t i = 0; i < size; i += 2) {
-      if constexpr (MulRandomWithEvens) {
-        coefficients[i >> 1] = coefficients_[i] * r;
-        coefficients[i >> 1] += coefficients_[i + 1];
-      } else {
-        coefficients[i >> 1] = coefficients_[i + 1] * r;
-        coefficients[i >> 1] += coefficients_[i];
-      }
+      coefficients[i >> 1] = coefficients_[i + 1] * r;
+      coefficients[i >> 1] += coefficients_[i];
     }
     if (size % 2 != 0) {
       coefficients[size >> 1] = coefficients_[size - 1];
-      if constexpr (MulRandomWithEvens) {
-        coefficients[size >> 1] *= r;
-      }
     }
     return UnivariateDenseCoefficients(std::move(coefficients), true);
   }
diff --git a/tachyon/math/polynomials/univariate/univariate_dense_polynomial_unittest.cc b/tachyon/math/polynomials/univariate/univariate_dense_polynomial_unittest.cc
index 178a6e77c..86778e17f 100644
--- a/tachyon/math/polynomials/univariate/univariate_dense_polynomial_unittest.cc
+++ b/tachyon/math/polynomials/univariate/univariate_dense_polynomial_unittest.cc
@@ -339,39 +339,21 @@ TEST_F(UnivariateDensePolynomialTest, EvaluateVanishingPolyByRoots) {
             poly.Evaluate(point));
 }
 
-TEST_F(UnivariateDensePolynomialTest, FoldEven) {
+TEST_F(UnivariateDensePolynomialTest, Fold) {
   Poly poly = Poly::Random(kMaxDegree);
   GF7 r = GF7::Random();
-  Poly folded = poly.Fold<true>(r);
-  EXPECT_EQ(folded, Poly(Coeffs({r * poly[0] + poly[1], r * poly[2] + poly[3],
-                                 r * poly[4] + poly[5]},
-                                true)));
-
-  GF7 r2 = GF7::Random();
-  Poly folded2 = folded.Fold<true>(r2);
-  EXPECT_EQ(folded2,
-            Poly(Coeffs({r2 * folded[0] + folded[1], r2 * folded[2]}, true)));
-
-  GF7 r3 = GF7::Random();
-  Poly folded3 = folded2.Fold<true>(r3);
-  EXPECT_EQ(folded3, Poly(Coeffs({r3 * folded2[0] + folded2[1]}, true)));
-}
-
-TEST_F(UnivariateDensePolynomialTest, FoldOdd) {
-  Poly poly = Poly::Random(kMaxDegree);
-  GF7 r = GF7::Random();
-  Poly folded = poly.Fold<false>(r);
+  Poly folded = poly.Fold(r);
   EXPECT_EQ(folded, Poly(Coeffs({poly[0] + r * poly[1], poly[2] + r * poly[3],
                                  poly[4] + r * poly[5]},
                                 true)));
 
   GF7 r2 = GF7::Random();
-  Poly folded2 = folded.Fold<false>(r2);
+  Poly folded2 = folded.Fold(r2);
   EXPECT_EQ(folded2,
             Poly(Coeffs({folded[0] + r2 * folded[1], folded[2]}, true)));
 
   GF7 r3 = GF7::Random();
-  Poly folded3 = folded2.Fold<false>(r3);
+  Poly folded3 = folded2.Fold(r3);
   EXPECT_EQ(folded3, Poly(Coeffs({folded2[0] + r3 * folded2[1]}, true)));
 }
 
diff --git a/tachyon/math/polynomials/univariate/univariate_polynomial.h b/tachyon/math/polynomials/univariate/univariate_polynomial.h
index 60fed66d2..b5ba41f28 100644
--- a/tachyon/math/polynomials/univariate/univariate_polynomial.h
+++ b/tachyon/math/polynomials/univariate/univariate_polynomial.h
@@ -150,12 +150,10 @@ class UnivariatePolynomial final
 
   // Return a polynomial where the original polynomial reduces its degree
   // by categorizing coefficients into even and odd degrees,
-  // multiplying either set of coefficients by a specified random field |r|,
+  // multiplying coefficients with odd degrees by a specified random field |r|,
   // and summing them together.
-  template <bool MulRandomWithEvens>
   constexpr UnivariatePolynomial Fold(const Field& r) const {
-    return UnivariatePolynomial(
-        coefficients_.template Fold<MulRandomWithEvens>(r));
+    return UnivariatePolynomial(coefficients_.Fold(r));
   }
 
   decltype(auto) ToSparse() const {
diff --git a/tachyon/math/polynomials/univariate/univariate_sparse_coefficients.h b/tachyon/math/polynomials/univariate/univariate_sparse_coefficients.h
index d84b9603e..4748e1d2f 100644
--- a/tachyon/math/polynomials/univariate/univariate_sparse_coefficients.h
+++ b/tachyon/math/polynomials/univariate/univariate_sparse_coefficients.h
@@ -227,36 +227,25 @@ class UnivariateSparseCoefficients {
 
   // Return coefficients where the original coefficients reduce their degree
   // by categorizing coefficients into even and odd degrees,
-  // multiplying either set of coefficients by a specified random field |r|,
+  // multiplying coefficients with odd degrees by a specified random field |r|,
   // and summing them together.
-  template <bool MulRandomWithEvens>
   constexpr UnivariateSparseCoefficients Fold(const Field& r) const {
     std::vector<Term> terms;
     terms.reserve(terms_.size() >> 1);
     bool r_is_zero = r.IsZero();
     for (const Term& term : terms_) {
       if (term.degree % 2 == 0) {
-        if constexpr (MulRandomWithEvens) {
-          if (!r_is_zero) {
-            terms.push_back({term.degree >> 1, term.coefficient * r});
-          }
-        } else {
-          terms.push_back({term.degree >> 1, term.coefficient});
-        }
+        terms.push_back({term.degree >> 1, term.coefficient});
       } else {
         if (!terms.empty() && terms.back().degree == (term.degree >> 1)) {
-          if constexpr (MulRandomWithEvens) {
-            terms.back() += term.coefficient;
-          } else if (!r_is_zero) {
+          if (!r_is_zero) {
             terms.back() += (term.coefficient * r);
           }
           if (terms.back().coefficient.IsZero()) {
             terms.pop_back();
           }
         } else {
-          if constexpr (MulRandomWithEvens) {
-            terms.push_back({term.degree >> 1, term.coefficient});
-          } else if (!r_is_zero) {
+          if (!r_is_zero) {
             terms.push_back({term.degree >> 1, term.coefficient * r});
           }
         }
diff --git a/tachyon/math/polynomials/univariate/univariate_sparse_polynomial_unittest.cc b/tachyon/math/polynomials/univariate/univariate_sparse_polynomial_unittest.cc
index fe37f5c13..b431327a2 100644
--- a/tachyon/math/polynomials/univariate/univariate_sparse_polynomial_unittest.cc
+++ b/tachyon/math/polynomials/univariate/univariate_sparse_polynomial_unittest.cc
@@ -369,39 +369,10 @@ TEST_F(UnivariateSparsePolynomialTest, EvaluateVanishingPolyByRoots) {
             poly.Evaluate(point));
 }
 
-TEST_F(UnivariateSparsePolynomialTest, FoldEven) {
+TEST_F(UnivariateSparsePolynomialTest, Fold) {
   Poly poly = Poly::Random(kMaxDegree);
   GF7 r = GF7::Random();
-  Poly folded = poly.Fold<true>(r);
-  std::vector<UnivariateTerm<GF7>> terms{{0, r * poly[0] + poly[1]},
-                                         {1, r * poly[2] + poly[3]},
-                                         {2, r * poly[4] + poly[5]}};
-  base::EraseIf(terms, [](const UnivariateTerm<GF7>& term) {
-    return term.coefficient.IsZero();
-  });
-  EXPECT_EQ(folded, Poly(Coeffs(std::move(terms))));
-
-  GF7 r2 = GF7::Random();
-  Poly folded2 = folded.Fold<true>(r2);
-  terms = {{0, r2 * folded[0] + folded[1]}, {1, r2 * folded[2]}};
-  base::EraseIf(terms, [](const UnivariateTerm<GF7>& term) {
-    return term.coefficient.IsZero();
-  });
-  EXPECT_EQ(folded2, Poly(Coeffs(std::move(terms))));
-
-  GF7 r3 = GF7::Random();
-  Poly folded3 = folded2.Fold<true>(r3);
-  terms = {{0, r3 * folded2[0] + folded2[1]}};
-  base::EraseIf(terms, [](const UnivariateTerm<GF7>& term) {
-    return term.coefficient.IsZero();
-  });
-  EXPECT_EQ(folded3, Poly(Coeffs(std::move(terms))));
-}
-
-TEST_F(UnivariateSparsePolynomialTest, FoldOdd) {
-  Poly poly = Poly::Random(kMaxDegree);
-  GF7 r = GF7::Random();
-  Poly folded = poly.Fold<false>(r);
+  Poly folded = poly.Fold(r);
   std::vector<UnivariateTerm<GF7>> terms{{0, poly[0] + r * poly[1]},
                                          {1, poly[2] + r * poly[3]},
                                          {2, poly[4] + r * poly[5]}};
@@ -411,7 +382,7 @@ TEST_F(UnivariateSparsePolynomialTest, FoldOdd) {
   EXPECT_EQ(folded, Poly(Coeffs(std::move(terms))));
 
   GF7 r2 = GF7::Random();
-  Poly folded2 = folded.Fold<false>(r2);
+  Poly folded2 = folded.Fold(r2);
   terms = {{0, folded[0] + r2 * folded[1]}, {1, folded[2]}};
   base::EraseIf(terms, [](const UnivariateTerm<GF7>& term) {
     return term.coefficient.IsZero();
@@ -419,7 +390,7 @@ TEST_F(UnivariateSparsePolynomialTest, FoldOdd) {
   EXPECT_EQ(folded2, Poly(Coeffs(std::move(terms))));
 
   GF7 r3 = GF7::Random();
-  Poly folded3 = folded2.Fold<false>(r3);
+  Poly folded3 = folded2.Fold(r3);
   terms = {{0, folded2[0] + r3 * folded2[1]}};
   base::EraseIf(terms, [](const UnivariateTerm<GF7>& term) {
     return term.coefficient.IsZero();
diff --git a/tachyon/node/math/elliptic_curves/short_weierstrass/BUILD.bazel b/tachyon/node/math/elliptic_curves/short_weierstrass/BUILD.bazel
index d0d5a37cb..5286313aa 100644
--- a/tachyon/node/math/elliptic_curves/short_weierstrass/BUILD.bazel
+++ b/tachyon/node/math/elliptic_curves/short_weierstrass/BUILD.bazel
@@ -11,7 +11,10 @@ tachyon_cc_library(
         "projective_point.h",
     ],
     deps = [
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:jacobian_point",
+        "//tachyon/math/geometry:point_xyzz",
+        "//tachyon/math/geometry:projective_point",
         "//tachyon/node/base:node_base",
     ],
 )
diff --git a/tachyon/node/math/elliptic_curves/short_weierstrass/jacobian_point.h b/tachyon/node/math/elliptic_curves/short_weierstrass/jacobian_point.h
index 37fe8292d..ebba65fd0 100644
--- a/tachyon/node/math/elliptic_curves/short_weierstrass/jacobian_point.h
+++ b/tachyon/node/math/elliptic_curves/short_weierstrass/jacobian_point.h
@@ -1,7 +1,7 @@
 #ifndef TACHYON_NODE_MATH_ELLIPTIC_CURVES_SHORT_WEIERSTRASS_JACOBIAN_POINT_H_
 #define TACHYON_NODE_MATH_ELLIPTIC_CURVES_SHORT_WEIERSTRASS_JACOBIAN_POINT_H_
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/node/base/node_module.h"
 
 namespace tachyon::node::math {
diff --git a/tachyon/node/math/elliptic_curves/short_weierstrass/point_xyzz.h b/tachyon/node/math/elliptic_curves/short_weierstrass/point_xyzz.h
index 7067c63c1..6b2b5c574 100644
--- a/tachyon/node/math/elliptic_curves/short_weierstrass/point_xyzz.h
+++ b/tachyon/node/math/elliptic_curves/short_weierstrass/point_xyzz.h
@@ -1,7 +1,7 @@
 #ifndef TACHYON_NODE_MATH_ELLIPTIC_CURVES_SHORT_WEIERSTRASS_POINT_XYZZ_H_
 #define TACHYON_NODE_MATH_ELLIPTIC_CURVES_SHORT_WEIERSTRASS_POINT_XYZZ_H_
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/node/base/node_module.h"
 
 namespace tachyon::node::math {
diff --git a/tachyon/node/math/elliptic_curves/short_weierstrass/projective_point.h b/tachyon/node/math/elliptic_curves/short_weierstrass/projective_point.h
index 1c9846979..abf788df4 100644
--- a/tachyon/node/math/elliptic_curves/short_weierstrass/projective_point.h
+++ b/tachyon/node/math/elliptic_curves/short_weierstrass/projective_point.h
@@ -1,7 +1,7 @@
 #ifndef TACHYON_NODE_MATH_ELLIPTIC_CURVES_SHORT_WEIERSTRASS_PROJECTIVE_POINT_H_
 #define TACHYON_NODE_MATH_ELLIPTIC_CURVES_SHORT_WEIERSTRASS_PROJECTIVE_POINT_H_
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/node/base/node_module.h"
 
 namespace tachyon::node::math {
diff --git a/tachyon/py/math/elliptic_curves/short_weierstrass/BUILD.bazel b/tachyon/py/math/elliptic_curves/short_weierstrass/BUILD.bazel
index 5f7cd4f0f..e87c46bcf 100644
--- a/tachyon/py/math/elliptic_curves/short_weierstrass/BUILD.bazel
+++ b/tachyon/py/math/elliptic_curves/short_weierstrass/BUILD.bazel
@@ -11,7 +11,10 @@ tachyon_pybind_library(
         "projective_point.h",
     ],
     deps = [
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:affine_point",
+        "//tachyon/math/geometry:jacobian_point",
+        "//tachyon/math/geometry:point_xyzz",
+        "//tachyon/math/geometry:projective_point",
         "//tachyon/py/base:pybind11",
     ],
 )
diff --git a/tachyon/py/math/elliptic_curves/short_weierstrass/jacobian_point.h b/tachyon/py/math/elliptic_curves/short_weierstrass/jacobian_point.h
index 87ac89569..583197ee7 100644
--- a/tachyon/py/math/elliptic_curves/short_weierstrass/jacobian_point.h
+++ b/tachyon/py/math/elliptic_curves/short_weierstrass/jacobian_point.h
@@ -5,7 +5,7 @@
 
 #include "pybind11/operators.h"
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/py/base/pybind11.h"
 
 namespace tachyon::py::math {
diff --git a/tachyon/py/math/elliptic_curves/short_weierstrass/point_xyzz.h b/tachyon/py/math/elliptic_curves/short_weierstrass/point_xyzz.h
index 92f9ba911..1c3915a4b 100644
--- a/tachyon/py/math/elliptic_curves/short_weierstrass/point_xyzz.h
+++ b/tachyon/py/math/elliptic_curves/short_weierstrass/point_xyzz.h
@@ -5,7 +5,7 @@
 
 #include "pybind11/operators.h"
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/py/base/pybind11.h"
 
 namespace tachyon::py::math {
diff --git a/tachyon/py/math/elliptic_curves/short_weierstrass/projective_point.h b/tachyon/py/math/elliptic_curves/short_weierstrass/projective_point.h
index 9f837ce64..10de12c58 100644
--- a/tachyon/py/math/elliptic_curves/short_weierstrass/projective_point.h
+++ b/tachyon/py/math/elliptic_curves/short_weierstrass/projective_point.h
@@ -5,7 +5,7 @@
 
 #include "pybind11/operators.h"
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/py/base/pybind11.h"
 
 namespace tachyon::py::math {
diff --git a/tachyon/zk/plonk/halo2/BUILD.bazel b/tachyon/zk/plonk/halo2/BUILD.bazel
index 2e17f21f8..4a5114df0 100644
--- a/tachyon/zk/plonk/halo2/BUILD.bazel
+++ b/tachyon/zk/plonk/halo2/BUILD.bazel
@@ -181,8 +181,8 @@ tachyon_cc_library(
     hdrs = ["proof_serializer.h"],
     deps = [
         "//tachyon/base/buffer",
-        "//tachyon/math/elliptic_curves:points",
         "//tachyon/math/finite_fields:prime_field_base",
+        "//tachyon/math/geometry:affine_point",
     ],
 )
 
diff --git a/tachyon/zk/plonk/halo2/proof_serializer.h b/tachyon/zk/plonk/halo2/proof_serializer.h
index 436dbca14..6f290b8df 100644
--- a/tachyon/zk/plonk/halo2/proof_serializer.h
+++ b/tachyon/zk/plonk/halo2/proof_serializer.h
@@ -11,8 +11,8 @@
 #include <utility>
 
 #include "tachyon/base/buffer/buffer.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
 #include "tachyon/math/finite_fields/prime_field_base.h"
+#include "tachyon/math/geometry/affine_point.h"
 
 namespace tachyon::zk::plonk::halo2 {
 
diff --git a/tachyon/zk/plonk/halo2/stringifiers/BUILD.bazel b/tachyon/zk/plonk/halo2/stringifiers/BUILD.bazel
index 1781cca71..8b111e2e8 100644
--- a/tachyon/zk/plonk/halo2/stringifiers/BUILD.bazel
+++ b/tachyon/zk/plonk/halo2/stringifiers/BUILD.bazel
@@ -110,7 +110,7 @@ tachyon_cc_library(
     hdrs = ["point_stringifier.h"],
     deps = [
         ":field_stringifier",
-        "//tachyon/math/elliptic_curves:points",
+        "//tachyon/math/geometry:affine_point",
     ],
 )
 
diff --git a/tachyon/zk/plonk/halo2/stringifiers/point_stringifier.h b/tachyon/zk/plonk/halo2/stringifiers/point_stringifier.h
index 4009a67cc..e5de9d08a 100644
--- a/tachyon/zk/plonk/halo2/stringifiers/point_stringifier.h
+++ b/tachyon/zk/plonk/halo2/stringifiers/point_stringifier.h
@@ -10,7 +10,7 @@
 #include <ostream>
 
 #include "tachyon/base/strings/rust_stringifier.h"
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 #include "tachyon/zk/plonk/halo2/stringifiers/field_stringifier.h"
 
 namespace tachyon::base::internal {
diff --git a/tachyon/zk/r1cs/groth16/BUILD.bazel b/tachyon/zk/r1cs/groth16/BUILD.bazel
index 11a613c52..a4214af9b 100644
--- a/tachyon/zk/r1cs/groth16/BUILD.bazel
+++ b/tachyon/zk/r1cs/groth16/BUILD.bazel
@@ -67,8 +67,8 @@ tachyon_cc_library(
     deps = [
         ":prepared_verifying_key",
         ":proof",
-        "//tachyon/math/elliptic_curves:points",
         "//tachyon/math/elliptic_curves/msm:variable_base_msm",
+        "//tachyon/math/geometry:point_conversions",
     ],
 )
 
diff --git a/tachyon/zk/r1cs/groth16/verify.h b/tachyon/zk/r1cs/groth16/verify.h
index 8bfa302c3..b88493afe 100644
--- a/tachyon/zk/r1cs/groth16/verify.h
+++ b/tachyon/zk/r1cs/groth16/verify.h
@@ -10,7 +10,7 @@
 
 #include "tachyon/math/elliptic_curves/msm/variable_base_msm.h"
 #include "tachyon/math/elliptic_curves/pairing/pairing.h"
-#include "tachyon/math/elliptic_curves/point_conversions.h"
+#include "tachyon/math/geometry/point_conversions.h"
 #include "tachyon/zk/r1cs/groth16/prepared_verifying_key.h"
 #include "tachyon/zk/r1cs/groth16/proof.h"
 
diff --git a/vendors/circom/circomlib/json/BUILD.bazel b/vendors/circom/circomlib/json/BUILD.bazel
index 2018c2d5e..bcca2a02c 100644
--- a/vendors/circom/circomlib/json/BUILD.bazel
+++ b/vendors/circom/circomlib/json/BUILD.bazel
@@ -8,6 +8,7 @@ tachyon_cc_library(
     deps = [
         ":json_converter_forward",
         ":points",
+        "@kroma_network_tachyon//tachyon/base:logging",
         "@kroma_network_tachyon//tachyon/zk/r1cs/groth16:proof",
     ],
 )
@@ -33,7 +34,7 @@ tachyon_cc_library(
     hdrs = ["points.h"],
     deps = [
         "@com_github_tencent_rapidjson//:rapidjson",
-        "@kroma_network_tachyon//tachyon/math/elliptic_curves:points",
+        "@kroma_network_tachyon//tachyon/math/geometry:affine_point",
     ],
 )
 
diff --git a/vendors/circom/circomlib/json/groth16_proof.h b/vendors/circom/circomlib/json/groth16_proof.h
index f40f0691b..896fa7ca3 100644
--- a/vendors/circom/circomlib/json/groth16_proof.h
+++ b/vendors/circom/circomlib/json/groth16_proof.h
@@ -3,6 +3,7 @@
 
 #include "circomlib/json/json_converter_forward.h"
 #include "circomlib/json/points.h"
+#include "tachyon/base/logging.h"
 #include "tachyon/zk/r1cs/groth16/proof.h"
 
 namespace tachyon::circom {
diff --git a/vendors/circom/circomlib/json/points.h b/vendors/circom/circomlib/json/points.h
index 2a5d431e2..fe0cd76d0 100644
--- a/vendors/circom/circomlib/json/points.h
+++ b/vendors/circom/circomlib/json/points.h
@@ -5,7 +5,7 @@
 
 #include "rapidjson/document.h"
 
-#include "tachyon/math/elliptic_curves/affine_point.h"
+#include "tachyon/math/geometry/affine_point.h"
 
 namespace tachyon::circom::internal {