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From a triangular mesh I have a 3D triangle T represented by 3D vertexes v0, v1 and v2. Each vertex has an associated UV coordinate into a common texture image, represented as uv0, uv1, uv2.

Furthermore there is 3D point P, that is guaranteed to be within the triangle T (coplanar to and within boundary of T).

My question is given point P, and triangle T, how does one go about calculating the UV coordinates of P?

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Here is some C++ code from javidx9 from his olc3DEngine library. Hope this helps.

void TexturedTriangle(  int x1, int y1, float u1, float v1, float w1,
                            int x2, int y2, float u2, float v2, float w2,
                            int x3, int y3, float u3, float v3, float w3,
        olcSprite *tex)
    {
        if (y2 < y1)
        {
            swap(y1, y2);
            swap(x1, x2);
            swap(u1, u2);
            swap(v1, v2);
            swap(w1, w2);
        }

        if (y3 < y1)
        {
            swap(y1, y3);
            swap(x1, x3);
            swap(u1, u3);
            swap(v1, v3);
            swap(w1, w3);
        }

        if (y3 < y2)
        {
            swap(y2, y3);
            swap(x2, x3);
            swap(u2, u3);
            swap(v2, v3);
            swap(w2, w3);
        }

        int dy1 = y2 - y1;
        int dx1 = x2 - x1;
        float dv1 = v2 - v1;
        float du1 = u2 - u1;
        float dw1 = w2 - w1;

        int dy2 = y3 - y1;
        int dx2 = x3 - x1;
        float dv2 = v3 - v1;
        float du2 = u3 - u1;
        float dw2 = w3 - w1;

        float tex_u, tex_v, tex_w;

        float dax_step = 0, dbx_step = 0,
            du1_step = 0, dv1_step = 0,
            du2_step = 0, dv2_step = 0,
            dw1_step=0, dw2_step=0;

        if (dy1) dax_step = dx1 / (float)abs(dy1);
        if (dy2) dbx_step = dx2 / (float)abs(dy2);

        if (dy1) du1_step = du1 / (float)abs(dy1);
        if (dy1) dv1_step = dv1 / (float)abs(dy1);
        if (dy1) dw1_step = dw1 / (float)abs(dy1);

        if (dy2) du2_step = du2 / (float)abs(dy2);
        if (dy2) dv2_step = dv2 / (float)abs(dy2);
        if (dy2) dw2_step = dw2 / (float)abs(dy2);

        if (dy1)
        {
            for (int i = y1; i <= y2; i++)
            {
                int ax = x1 + (float)(i - y1) * dax_step;
                int bx = x1 + (float)(i - y1) * dbx_step;

                float tex_su = u1 + (float)(i - y1) * du1_step;
                float tex_sv = v1 + (float)(i - y1) * dv1_step;
                float tex_sw = w1 + (float)(i - y1) * dw1_step;

                float tex_eu = u1 + (float)(i - y1) * du2_step;
                float tex_ev = v1 + (float)(i - y1) * dv2_step;
                float tex_ew = w1 + (float)(i - y1) * dw2_step;

                if (ax > bx)
                {
                    swap(ax, bx);
                    swap(tex_su, tex_eu);
                    swap(tex_sv, tex_ev);
                    swap(tex_sw, tex_ew);
                }

                tex_u = tex_su;
                tex_v = tex_sv;
                tex_w = tex_sw;

                float tstep = 1.0f / ((float)(bx - ax));
                float t = 0.0f;

                for (int j = ax; j < bx; j++)
                {
                    tex_u = (1.0f - t) * tex_su + t * tex_eu;
                    tex_v = (1.0f - t) * tex_sv + t * tex_ev;
                    tex_w = (1.0f - t) * tex_sw + t * tex_ew;
                    if (tex_w > pDepthBuffer[i*ScreenWidth() + j])
                    {
                        Draw(j, i, tex->SampleGlyph(tex_u / tex_w, tex_v / tex_w), tex->SampleColour(tex_u / tex_w, tex_v / tex_w));
                        pDepthBuffer[i*ScreenWidth() + j] = tex_w;
                    }
                    t += tstep;
                }

            }
        }

        dy1 = y3 - y2;
        dx1 = x3 - x2;
        dv1 = v3 - v2;
        du1 = u3 - u2;
        dw1 = w3 - w2;

        if (dy1) dax_step = dx1 / (float)abs(dy1);
        if (dy2) dbx_step = dx2 / (float)abs(dy2);

        du1_step = 0, dv1_step = 0;
        if (dy1) du1_step = du1 / (float)abs(dy1);
        if (dy1) dv1_step = dv1 / (float)abs(dy1);
        if (dy1) dw1_step = dw1 / (float)abs(dy1);

        if (dy1)
        {
            for (int i = y2; i <= y3; i++)
            {
                int ax = x2 + (float)(i - y2) * dax_step;
                int bx = x1 + (float)(i - y1) * dbx_step;

                float tex_su = u2 + (float)(i - y2) * du1_step;
                float tex_sv = v2 + (float)(i - y2) * dv1_step;
                float tex_sw = w2 + (float)(i - y2) * dw1_step;

                float tex_eu = u1 + (float)(i - y1) * du2_step;
                float tex_ev = v1 + (float)(i - y1) * dv2_step;
                float tex_ew = w1 + (float)(i - y1) * dw2_step;

                if (ax > bx)
                {
                    swap(ax, bx);
                    swap(tex_su, tex_eu);
                    swap(tex_sv, tex_ev);
                    swap(tex_sw, tex_ew);
                }

                tex_u = tex_su;
                tex_v = tex_sv;
                tex_w = tex_sw;

                float tstep = 1.0f / ((float)(bx - ax));
                float t = 0.0f;

                for (int j = ax; j < bx; j++)
                {
                    tex_u = (1.0f - t) * tex_su + t * tex_eu;
                    tex_v = (1.0f - t) * tex_sv + t * tex_ev;
                    tex_w = (1.0f - t) * tex_sw + t * tex_ew;

                    if (tex_w > pDepthBuffer[i*ScreenWidth() + j])
                    {
                        Draw(j, i, tex->SampleGlyph(tex_u / tex_w, tex_v / tex_w), tex->SampleColour(tex_u / tex_w, tex_v / tex_w));
                        pDepthBuffer[i*ScreenWidth() + j] = tex_w;
                    }
                    t += tstep;
                }
            }   
        }       
    }
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