All vector variables are marked bold.

Given:

R₀ - start point of the ray,

Dir - direction of the ray,

|Dir| = 1 - direction is normalized to be of length 1

A, B, C - the triangle points

Specific point P at ray may be expressed as:

P = R₀ + t * Dir

From another side specific P may be expressed in terms of triangle. Let's introduce new coordinate system (U, V, W, A), where

U = B - A, first axis

V = C - A, second eaxis

W = U x V, third axis is calculated to be prependicular to both U and V. Dot product makes the job.

A, is our central point

Then point P in general case is equal:

P = U * u + V * v + W * w + A,

So what we need to find the intersection is just find these u, v, w

As our intersection must be in triangle plane we can say that

w = 0

We need to find u, v values. They may be also used as point to triangle texture material.

Let's introduce vector D which will be our point P but relative to our new coordinate system:

D = P - A

Or

D_x = U_x * u + V_x * v + W_x * w

D_y = U_y * u + V_y * v + W_y * w

D_z = U_z * u + V_z * v + W_z * w

These linear equations can be solved as

u = det_u / det

v = det_v / det

w = det_w / det

Or

det_u = D * (V x W)

det_v = D * (W x U)

det_w = D * (U x V)= D * W

u = det_u / det = D * (V x W) / det = D * T_u

v = det_v / det = D * (W x U) / det = D * T_v

w = det_w / det = D * (U x V) / det = D * T_w

w = 0

D * T_w = 0

(P - A) * T_w = 0

P * T_w = A * T_w

(R₀ + t * Dir) * T_w = A * T_w

R₀ * T_w + t * Dir * T_w = A * T_w

t * Dir * T_w = A * T_w - R₀ * T_w

t * Dir * T_w = (A - R₀) * T_w

t = (A - R₀) * T_w / (Dir * T_w)