Fix Triangle2d/Triangle3d interior sampling to correctly follow triangle

[mweatherley]

Objective

When I wrote #12747 I neglected to translate random samples from triangles back to the point where they originated, so they would be sampled near the origin instead of at the actual triangle location.

Solution

Translate by the first vertex location so that the samples follow the actual triangle.

[bushrat011899]

I agree this solves the bug, not sure how I missed it in the initial review! Just as an aside, this function can be made a lot simpler now that I'm looking at it more closely:

/// Interior sampling for triangles which doesn't depend on the ambient dimension.
fn sample_triangle_interior<P: NormedVectorSpace, R: Rng + ?Sized>(
    [a, b, c]: [P; 3],
    rng: &mut R,
) -> P {
    // Generate random points on a parallelipiped and reflect so that
    // we can use the points that lie outside the triangle
    let u = rng.gen_range(0.0..=1.0);
    let v = rng.gen_range(0.0..=1.0);

    a * (1. - u - v).abs() + b * u + c * v;
}

This removes 1 P-space vector addition and eliminates the branch, and also includes the fix you've developed. Even without this suggestion, the fix you've provided makes sense and I approve!

[mweatherley]

I agree this solves the bug, not sure how I missed it in the initial review! Just as an aside, this function can be made a lot simpler now that I'm looking at it more closely:

/// Interior sampling for triangles which doesn't depend on the ambient dimension.
fn sample_triangle_interior<P: NormedVectorSpace, R: Rng + ?Sized>(
    [a, b, c]: [P; 3],
    rng: &mut R,
) -> P {
    // Generate random points on a parallelipiped and reflect so that
    // we can use the points that lie outside the triangle
    let u = rng.gen_range(0.0..=1.0);
    let v = rng.gen_range(0.0..=1.0);

    a * (1. - u - v).abs() + b * u + c * v;
}

This removes 1 P-space vector addition and eliminates the branch, and also includes the fix you've developed. Even without this suggestion, the fix you've provided makes sense and I approve!

I don't think this is right; when the inside of the absolute value is negative, the sum of coefficients is 2u + 2v - 1, when it should always be 1.

[bushrat011899]

Yes you're correct, I gotta stop trying to do maths at midnight. I'd neglected the change in the coefficients of b and c.

/// Interior sampling for triangles which doesn't depend on the ambient dimension.
fn sample_triangle_interior<P: NormedVectorSpace, R: Rng + ?Sized>(
    [a, b, c]: [P; 3],
    rng: &mut R,
) -> P {
    // Generate random points on a parallelipiped
    let u = rng.gen_range(0.0..=1.0);
    let v = rng.gen_range(0.0..=1.0);

    // Reflect so that we can use the points that lie outside the triangle
    let (u, v) = if u + v > 1. {
        (1. - u, 1. - v)
    } else {
        (u, v)
    };

    // Ensure coefficients sum to 1
    let t = 1. - u - v;

    a * t + b * u + c * v;
}

Still removes the extra vector operation, but definitely more of a style choice than anything else now.

[alice-i-cecile]

I'd like to see tests that our sampling methods always return values inside of the shapes, but I won't block on it here.