Here is one easy way to understand why this is equivalent to the previous definition of alpha, beta, and gamma (ratio of lengths). Let's take for an example the expression for alpha. The area of A_A is equal to base * height / 2, which is BC * L_BC(A_A) / 2 when dropping the height down from A_A to BC. The area of A_A + A_B + A_C is just the area of the entire triangle, which is equal to base * height / 2 = BC * L_BC(A) / 2. Taking the ratio of these expressions of these areas simplifies out to L_BC(A_A) / L_BC(A), which is equal to alpha (recall that from the proportional definition of alpha, the altitude of triangle ABC when dropped from vertex A is represented by (1 - alpha) + alpha = 1)
abhim00
When implementing the project do we use these weights alpha beta gamma to compute the new interpolated color.
Here is one easy way to understand why this is equivalent to the previous definition of alpha, beta, and gamma (ratio of lengths). Let's take for an example the expression for alpha. The area of A_A is equal to base * height / 2, which is BC * L_BC(A_A) / 2 when dropping the height down from A_A to BC. The area of A_A + A_B + A_C is just the area of the entire triangle, which is equal to base * height / 2 = BC * L_BC(A) / 2. Taking the ratio of these expressions of these areas simplifies out to L_BC(A_A) / L_BC(A), which is equal to alpha (recall that from the proportional definition of alpha, the altitude of triangle ABC when dropped from vertex A is represented by (1 - alpha) + alpha = 1)
When implementing the project do we use these weights alpha beta gamma to compute the new interpolated color.