diff --git a/src/arithmetic/curves.rs b/src/arithmetic/curves.rs
index ef1ba4fb..21da4f74 100644
--- a/src/arithmetic/curves.rs
+++ b/src/arithmetic/curves.rs
@@ -84,9 +84,12 @@ pub trait Curve:
     /// # Example
     ///
     /// ```
-    /// use halo2::arithmetic::{Curve, CurveAffine};
-    /// fn pedersen_commitment<C: CurveAffine>(x: C::Scalar, r: C::Scalar) -> C {
-    ///     let hasher = C::Curve::hash_to_curve("z.cash:example_pedersen_commitment");
+    /// use halo2::arithmetic::Curve;
+    /// fn pedersen_commitment<C: Curve>(
+    ///     x: <C as Curve>::Scalar,
+    ///     r: <C as Curve>::Scalar,
+    /// ) -> C::Affine {
+    ///     let hasher = C::hash_to_curve("z.cash:example_pedersen_commitment");
     ///     let g = hasher(b"g");
     ///     let h = hasher(b"h");
     ///     (g * x + &(h * r)).to_affine()
diff --git a/src/pasta/curves.rs b/src/pasta/curves.rs
index 9a2b9db1..fe266ccd 100644
--- a/src/pasta/curves.rs
+++ b/src/pasta/curves.rs
@@ -11,7 +11,7 @@ use super::{Fp, Fq};
 use crate::arithmetic::{Curve, CurveAffine, FieldExt, Group};
 
 macro_rules! new_curve_impl {
-    (($($privacy:tt)*), $name:ident, $name_affine:ident, $iso_affine:ident, $base:ident, $scalar:ident, $blake2b_personalization:literal,
+    (($($privacy:tt)*), $name:ident, $name_affine:ident, $iso:ident, $base:ident, $scalar:ident, $blake2b_personalization:literal,
      $curve_id:literal, $a_raw:expr, $b_raw:expr, $curve_type:ident) => {
         /// Represents a point in the projective coordinate space.
         #[derive(Copy, Clone, Debug)]
@@ -85,7 +85,7 @@ macro_rules! new_curve_impl {
                 $name_affine::conditional_select(&tmp, &$name_affine::identity(), zinv.ct_is_zero())
             }
 
-            impl_projective_curve_ext!($name, $name_affine, $iso_affine, $base, $curve_type);
+            impl_projective_curve_ext!($name, $name_affine, $iso, $base, $curve_type);
 
             fn a() -> Self::Base {
                 $name::curve_constant_a()
@@ -759,29 +759,26 @@ macro_rules! impl_projective_curve_specific {
 }
 
 macro_rules! impl_projective_curve_ext {
-    ($name:ident, $name_affine:ident, $iso_affine:ident, $base:ident, special_a0_b5) => {
+    ($name:ident, $name_affine:ident, $iso:ident, $base:ident, special_a0_b5) => {
         fn hash_to_curve<'a>(domain_prefix: &'a str) -> Box<dyn Fn(&[u8]) -> Self + 'a> {
             use super::hashtocurve;
 
             Box::new(move |message| {
                 let mut us = [Field::zero(); 2];
                 hashtocurve::hash_to_field($name_affine::CURVE_ID, domain_prefix, message, &mut us);
-                let q0 = hashtocurve::map_to_curve_simple_swu::<$base, $name_affine, $iso_affine>(
+                let q0 = hashtocurve::map_to_curve_simple_swu::<$base, $name, $iso>(
                     &us[0],
                     $name::THETA,
                     $name::Z,
                 );
-                let q1 = hashtocurve::map_to_curve_simple_swu::<$base, $name_affine, $iso_affine>(
+                let q1 = hashtocurve::map_to_curve_simple_swu::<$base, $name, $iso>(
                     &us[1],
                     $name::THETA,
                     $name::Z,
                 );
                 let r = q0 + &q1;
                 debug_assert!(bool::from(r.is_on_curve()));
-                hashtocurve::iso_map::<$base, $name_affine, $iso_affine>(
-                    &r,
-                    &$name::ISOGENY_CONSTANTS,
-                )
+                hashtocurve::iso_map::<$base, $name, $iso>(&r, &$name::ISOGENY_CONSTANTS)
             })
         }
 
@@ -795,7 +792,7 @@ macro_rules! impl_projective_curve_ext {
             }
         }
     };
-    ($name:ident, $name_affine:ident, $iso_affine:ident, $base:ident, general) => {
+    ($name:ident, $name_affine:ident, $iso:ident, $base:ident, general) => {
         /// Unimplemented: hashing to this curve is not supported
         fn hash_to_curve<'a>(_domain_prefix: &'a str) -> Box<dyn Fn(&[u8]) -> Self + 'a> {
             unimplemented!()
@@ -835,7 +832,7 @@ new_curve_impl!(
     (pub),
     Ep,
     EpAffine,
-    IsoEpAffine,
+    IsoEp,
     Fp,
     Fq,
     b"halo2_____pallas",
@@ -848,7 +845,7 @@ new_curve_impl!(
     (pub),
     Eq,
     EqAffine,
-    IsoEqAffine,
+    IsoEq,
     Fq,
     Fp,
     b"halo2______vesta",
@@ -861,7 +858,7 @@ new_curve_impl!(
     (pub(crate)),
     IsoEp,
     IsoEpAffine,
-    EpAffine,
+    Ep,
     Fp,
     Fq,
     b"halo2_iso_pallas",
@@ -879,7 +876,7 @@ new_curve_impl!(
     (pub(crate)),
     IsoEq,
     IsoEqAffine,
-    EqAffine,
+    Eq,
     Fq,
     Fp,
     b"halo2__iso_vesta",
diff --git a/src/pasta/hashtocurve.rs b/src/pasta/hashtocurve.rs
index c3154557..d3acfe81 100644
--- a/src/pasta/hashtocurve.rs
+++ b/src/pasta/hashtocurve.rs
@@ -3,7 +3,7 @@
 
 use subtle::ConstantTimeEq;
 
-use crate::arithmetic::{Curve, CurveAffine, FieldExt};
+use crate::arithmetic::{Curve, FieldExt};
 
 /// Hashes over a message and writes the output to all of `buf`.
 pub fn hash_to_field<F: FieldExt>(
@@ -72,10 +72,10 @@ pub fn hash_to_field<F: FieldExt>(
 }
 
 /// Implements a degree 3 isogeny map.
-pub fn iso_map<F: FieldExt, C: CurveAffine<Base = F>, I: CurveAffine<Base = F>>(
-    p: &I::Curve,
+pub fn iso_map<F: FieldExt, C: Curve<Base = F>, I: Curve<Base = F>>(
+    p: &I,
     iso: &[C::Base; 13],
-) -> C::Curve {
+) -> C {
     // The input and output are in Jacobian coordinates, using the method
     // in "Avoiding inversions" [WB2019, section 4.3].
 
@@ -96,14 +96,14 @@ pub fn iso_map<F: FieldExt, C: CurveAffine<Base = F>, I: CurveAffine<Base = F>>(
     let xo = num_x * div_y * zo;
     let yo = num_y * div_x * zo.square();
 
-    C::Curve::new_jacobian(xo, yo, zo).unwrap()
+    C::new_jacobian(xo, yo, zo).unwrap()
 }
 
-pub fn map_to_curve_simple_swu<F: FieldExt, C: CurveAffine<Base = F>, I: CurveAffine<Base = F>>(
+pub fn map_to_curve_simple_swu<F: FieldExt, C: Curve<Base = F>, I: Curve<Base = F>>(
     u: &F,
     theta: F,
     z: F,
-) -> I::Curve {
+) -> I {
     // 1. tv1 = inv0(Z^2 * u^4 + Z * u^2)
     // 2. x1 = (-B / A) * (1 + tv1)
     // 3. If tv1 == 0, set x1 = B / (Z * A)
@@ -171,5 +171,5 @@ pub fn map_to_curve_simple_swu<F: FieldExt, C: CurveAffine<Base = F>, I: CurveAf
         (u.get_lower_32() % 2).ct_eq(&(y.get_lower_32() % 2)),
     );
 
-    I::Curve::new_jacobian(num_x * div, y * div3, div).unwrap()
+    I::new_jacobian(num_x * div, y * div3, div).unwrap()
 }
diff --git a/src/pasta/pallas.rs b/src/pasta/pallas.rs
index ea5de3fe..729b264a 100644
--- a/src/pasta/pallas.rs
+++ b/src/pasta/pallas.rs
@@ -42,8 +42,7 @@ fn test_iso_map() {
         ]),
     )
     .unwrap();
-    let p =
-        super::hashtocurve::iso_map::<_, Affine, super::IsoEpAffine>(&r, &Ep::ISOGENY_CONSTANTS);
+    let p = super::hashtocurve::iso_map::<_, Point, super::IsoEp>(&r, &Ep::ISOGENY_CONSTANTS);
     let (x, y, z) = p.jacobian_coordinates();
     assert!(
         format!("{:?}", x) == "0x318cc15f281662b3f26d0175cab97b924870c837879cac647e877be51a85e898"
@@ -58,8 +57,7 @@ fn test_iso_map() {
     // check that iso_map([2] r) = [2] iso_map(r)
     let r2 = r.double();
     assert!(bool::from(r2.is_on_curve()));
-    let p2 =
-        super::hashtocurve::iso_map::<_, Affine, super::IsoEpAffine>(&r2, &Ep::ISOGENY_CONSTANTS);
+    let p2 = super::hashtocurve::iso_map::<_, Point, super::IsoEp>(&r2, &Ep::ISOGENY_CONSTANTS);
     assert!(bool::from(p2.is_on_curve()));
     assert!(p2 == p.double());
 }
@@ -92,8 +90,7 @@ fn test_iso_map_identity() {
     let r = (r * -Fq::one()) + r;
     assert!(bool::from(r.is_on_curve()));
     assert!(bool::from(r.is_identity()));
-    let p =
-        super::hashtocurve::iso_map::<_, Affine, super::IsoEpAffine>(&r, &Ep::ISOGENY_CONSTANTS);
+    let p = super::hashtocurve::iso_map::<_, Point, super::IsoEp>(&r, &Ep::ISOGENY_CONSTANTS);
     assert!(bool::from(p.is_on_curve()));
     assert!(bool::from(p.is_identity()));
 }
@@ -101,12 +98,11 @@ fn test_iso_map_identity() {
 #[test]
 fn test_map_to_curve_simple_swu() {
     use crate::arithmetic::Curve;
-    use crate::pasta::curves::{IsoEp, IsoEpAffine};
+    use crate::pasta::curves::IsoEp;
     use crate::pasta::hashtocurve::map_to_curve_simple_swu;
 
     // The zero input is a special case.
-    let p: IsoEp =
-        map_to_curve_simple_swu::<Fp, EpAffine, IsoEpAffine>(&Fp::zero(), Ep::THETA, Ep::Z);
+    let p: IsoEp = map_to_curve_simple_swu::<Fp, Ep, IsoEp>(&Fp::zero(), Ep::THETA, Ep::Z);
     let (x, y, z) = p.jacobian_coordinates();
     println!("{:?}", p);
     assert!(
@@ -119,8 +115,7 @@ fn test_map_to_curve_simple_swu() {
         format!("{:?}", z) == "0x054b3ba10416dc104157b1318534a19d5d115472da7d746f8a5f250cd8cdef36"
     );
 
-    let p: IsoEp =
-        map_to_curve_simple_swu::<Fp, EpAffine, IsoEpAffine>(&Fp::one(), Ep::THETA, Ep::Z);
+    let p: IsoEp = map_to_curve_simple_swu::<Fp, Ep, IsoEp>(&Fp::one(), Ep::THETA, Ep::Z);
     let (x, y, z) = p.jacobian_coordinates();
     println!("{:?}", p);
     assert!(
diff --git a/src/pasta/vesta.rs b/src/pasta/vesta.rs
index be387c26..d41d60f4 100644
--- a/src/pasta/vesta.rs
+++ b/src/pasta/vesta.rs
@@ -17,12 +17,11 @@ pub type Affine = EqAffine;
 #[test]
 fn test_map_to_curve_simple_swu() {
     use crate::arithmetic::Curve;
-    use crate::pasta::curves::{IsoEq, IsoEqAffine};
+    use crate::pasta::curves::IsoEq;
     use crate::pasta::hashtocurve::map_to_curve_simple_swu;
 
     // The zero input is a special case.
-    let p: IsoEq =
-        map_to_curve_simple_swu::<Fq, EqAffine, IsoEqAffine>(&Fq::zero(), Eq::THETA, Eq::Z);
+    let p: IsoEq = map_to_curve_simple_swu::<Fq, Eq, IsoEq>(&Fq::zero(), Eq::THETA, Eq::Z);
     let (x, y, z) = p.jacobian_coordinates();
     println!("{:?}", p);
     assert!(
@@ -35,8 +34,7 @@ fn test_map_to_curve_simple_swu() {
         format!("{:?}", z) == "0x0b851e9e579403a76df1100f556e1f226e5656bdf38f3bf8601d8a3a9a15890b"
     );
 
-    let p: IsoEq =
-        map_to_curve_simple_swu::<Fq, EqAffine, IsoEqAffine>(&Fq::one(), Eq::THETA, Eq::Z);
+    let p: IsoEq = map_to_curve_simple_swu::<Fq, Eq, IsoEq>(&Fq::one(), Eq::THETA, Eq::Z);
     let (x, y, z) = p.jacobian_coordinates();
     println!("{:?}", p);
     assert!(