diff --git a/src/base/construction.rs b/src/base/construction.rs
index b606534c..f8e9e5f8 100644
--- a/src/base/construction.rs
+++ b/src/base/construction.rs
@@ -284,7 +284,8 @@ where
     where
         Standard: Distribution<N>,
     {
-        Self::from_fn_generic(nrows, ncols, |_, _| rand::random())
+        let mut rng = rand::thread_rng();
+        Self::from_fn_generic(nrows, ncols, |_, _| rng.gen())
     }
 
     /// Creates a matrix filled with random values from the given distribution.
diff --git a/src/geometry/quaternion_construction.rs b/src/geometry/quaternion_construction.rs
index 89911b91..625342a8 100644
--- a/src/geometry/quaternion_construction.rs
+++ b/src/geometry/quaternion_construction.rs
@@ -7,7 +7,7 @@ use quickcheck::{Arbitrary, Gen};
 
 #[cfg(feature = "rand-no-std")]
 use rand::{
-    distributions::{Distribution, OpenClosed01, Standard},
+    distributions::{Distribution, OpenClosed01, Standard, Uniform, uniform::SampleUniform},
     Rng,
 };
 
@@ -855,6 +855,7 @@ impl<N: SimdRealField> Distribution<UnitQuaternion<N>> for Standard
 where
     N::Element: SimdRealField,
     OpenClosed01: Distribution<N>,
+    N: SampleUniform,
 {
     /// Generate a uniformly distributed random rotation quaternion.
     #[inline]
@@ -863,10 +864,9 @@ where
         // Uniform random rotations.
         // In D. Kirk, editor, Graphics Gems III, pages 124-132. Academic, New York, 1992.
         let x0 = rng.sample(OpenClosed01);
-        let x1 = rng.sample(OpenClosed01);
-        let x2 = rng.sample(OpenClosed01);
-        let theta1 = N::simd_two_pi() * x1;
-        let theta2 = N::simd_two_pi() * x2;
+        let twopi = Uniform::new(N::zero(), N::simd_two_pi());
+        let theta1 = rng.sample(&twopi);
+        let theta2 = rng.sample(&twopi);
         let s1 = theta1.simd_sin();
         let c1 = theta1.simd_cos();
         let s2 = theta2.simd_sin();
diff --git a/src/geometry/rotation_specialization.rs b/src/geometry/rotation_specialization.rs
index 95e19da1..fe431804 100644
--- a/src/geometry/rotation_specialization.rs
+++ b/src/geometry/rotation_specialization.rs
@@ -7,7 +7,7 @@ use num::Zero;
 
 #[cfg(feature = "rand-no-std")]
 use rand::{
-    distributions::{Distribution, OpenClosed01, Standard},
+    distributions::{Distribution, OpenClosed01, Standard, Uniform, uniform::SampleUniform},
     Rng,
 };
 
@@ -265,12 +265,13 @@ impl<N: SimdRealField> Rotation2<N> {
 impl<N: SimdRealField> Distribution<Rotation2<N>> for Standard
 where
     N::Element: SimdRealField,
-    OpenClosed01: Distribution<N>,
+    N: SampleUniform,
 {
     /// Generate a uniformly distributed random rotation.
     #[inline]
     fn sample<'a, R: Rng + ?Sized>(&self, rng: &'a mut R) -> Rotation2<N> {
-        Rotation2::new(rng.sample(OpenClosed01) * N::simd_two_pi())
+        let twopi = Uniform::new(N::zero(), N::simd_two_pi());
+        Rotation2::new(rng.sample(twopi))
     }
 }
 
@@ -923,6 +924,7 @@ impl<N: SimdRealField> Distribution<Rotation3<N>> for Standard
 where
     N::Element: SimdRealField,
     OpenClosed01: Distribution<N>,
+    N: SampleUniform,
 {
     /// Generate a uniformly distributed random rotation.
     #[inline]
@@ -932,7 +934,8 @@ where
         // In D. Kirk, editor, Graphics Gems III, pages 117-120. Academic, New York, 1992.
 
         // Compute a random rotation around Z
-        let theta = N::simd_two_pi() * rng.sample(OpenClosed01);
+        let twopi = Uniform::new(N::zero(), N::simd_two_pi());
+        let theta = rng.sample(&twopi);
         let (ts, tc) = theta.simd_sin_cos();
         let a = MatrixN::<N, U3>::new(
             tc,
@@ -947,7 +950,7 @@ where
         );
 
         // Compute a random rotation *of* Z
-        let phi = N::simd_two_pi() * rng.sample(OpenClosed01);
+        let phi = rng.sample(&twopi);
         let z = rng.sample(OpenClosed01);
         let (ps, pc) = phi.simd_sin_cos();
         let sqrt_z = z.simd_sqrt();
diff --git a/src/geometry/unit_complex_construction.rs b/src/geometry/unit_complex_construction.rs
index 9a24a57d..fc1ff131 100644
--- a/src/geometry/unit_complex_construction.rs
+++ b/src/geometry/unit_complex_construction.rs
@@ -3,7 +3,7 @@ use quickcheck::{Arbitrary, Gen};
 
 #[cfg(feature = "rand-no-std")]
 use rand::{
-    distributions::{Distribution, OpenClosed01, Standard},
+    distributions::{Distribution, Uniform, uniform::SampleUniform, Standard},
     Rng,
 };
 
@@ -401,12 +401,13 @@ where
 impl<N: SimdRealField> Distribution<UnitComplex<N>> for Standard
 where
     N::Element: SimdRealField,
-    OpenClosed01: Distribution<N>,
+    N: SampleUniform,
 {
     /// Generate a uniformly distributed random `UnitComplex`.
     #[inline]
     fn sample<'a, R: Rng + ?Sized>(&self, rng: &mut R) -> UnitComplex<N> {
-        UnitComplex::from_angle(rng.sample(OpenClosed01) * N::simd_two_pi())
+        let twopi = Uniform::new(N::zero(), N::simd_two_pi());
+        UnitComplex::from_angle(rng.sample(twopi))
     }
 }