Project point on 2D triangle back into 3D?

Question

I'm not sure what to search, so I haven't been able to find what I need.

Say I have a 3D triangle with points [0, 1, 1], [1, 0.5, 0.5], [0, 0, 0]. I discard the Z component to create a 2D triangle with points [0, 1], [1, 0.5], [0, 0]. (I think this is an orthographic projection?) Through an unimportant process I find some 2D point that lies within the 2D triangle, say [0.5, 0.5].

How do I take that 2D point and find what its Z value should be to have it lie on the plane formed by the original 3D triangle?

Answers (or dupe links!) that describe maths through code rather than mathematical symbols would be greatly appreciated; I struggle to read the types of answers you get on Math.SE.

Answer 1

Expanding on @Spektre's excellent answer, this was how I implemented a working solution. I'm working with Unity, so I used Ivan Kutskir's awesome lightweight C# matrix class to handle the matrix maths. There's probably faster/cleaner ways to do this but this was very easy and works correctly.

Obviously you have to ensure that when you discard the Z axis, you don't end up with a degenerate triangle.

// tri is a 3D triangle with points p0, p1 and p2
// point is a 2D point within that triangle, assuming the Z axis is discarded

/*
Equivalent to this part of @Spektre's answer:

| u |           | (q1.x - q0.x) , (q2.x - q0.x) , q0.x |   | q.x |
| v | = inverse | (q1.y - q0.y) , (q2.y - q0.y) , q0.y | * | q.y |
| 1 |           |       0       ,       0       ,   1  |   |  1  |
*/

Matrix m1 = new Matrix(3, 3);
Matrix m2 = new Matrix(3, 1);
m1[0, 0] = tri.p1.x - tri.p0.x;
m1[0, 1] = tri.p2.x - tri.p0.x;
m1[0, 2] = tri.p0.x;
m1[1, 0] = tri.p1.y - tri.p0.y;
m1[1, 1] = tri.p2.y - tri.p0.y;
m1[1, 2] = tri.p0.y;
m1[2, 0] = 0;
m1[2, 1] = 0;
m1[2, 2] = 1;
m2[0, 0] = point.x;
m2[1, 0] = point.y;
m2[2, 0] = 1;
Matrix mResult = m1.Invert() * m2;
float u = (float)mResult[0, 0];
float v = (float)mResult[1, 0];

/*
Equivalent to this part of @Spektre's answer:

p.x = p0.x + (p1.x - p0.x) * u + (p2.x - p0.x) * v
p.y = p0.y + (p1.y - p0.y) * u + (p2.y - p0.y) * v
p.z = p0.z + (p1.z - p0.z) * u + (p2.z - p0.z) * v
*/

float newX = tri.p0.x + (tri.p1.x - tri.p0.x) * u + (tri.p2.x - tri.p0.x) * v;
float newY = tri.p0.y + (tri.p1.y - tri.p0.y) * u + (tri.p2.y - tri.p0.y) * v;
float newZ = tri.p0.z + (tri.p1.z - tri.p0.z) * u + (tri.p2.z - tri.p0.z) * v;
Vector3 newPoint = new Vector3(newX, newY, newZ);

Alternatively, you can achieve the same result without the matrix (though this may be a less robust method, I'm not sure). To calculate the barycentric coordinates I used this implementation, but the accepted answer also works.

// tri is a 3D triangle with points p0, p1 and p2
// point is a 2D point within that triangle, assuming the Z axis is discarded

// Find the barycentric coords for the chosen 2D point...
float u, v, w = 0;
Barycentric2D(point, new Vector2(tri.p0.x, tri.p0.y), new Vector2(tri.p1.x, tri.p1.y), new Vector2(tri.p2.x, tri.p2.y), out u, out v, out w);

// ...and then find what the Z value would be for those barycentric coords in 3D
float newZ = tri.p0.z * u + tri.p1.z * v + tri.p2.z * w;
Vector3 newPoint = new Vector3(point.x, point.y, newZ);

// https://gamedev.stackexchange.com/a/63203/48697
void Barycentric2D(Vector2 p, Vector2 a, Vector2 b, Vector2 c, out float u, out float v, out float w)
{
    Vector2 v0 = b - a;
    Vector2 v1 = c - a;
    Vector2 v2 = p - a;
    float den = v0.x * v1.y - v1.x * v0.y;
    v = (v2.x * v1.y - v1.x * v2.y) / den;
    w = (v0.x * v2.y - v2.x * v0.y) / den;
    u = 1.0f - v - w;
}