The given question is as follows:
Consider a line segment AB parallel to the Z axis with end points A[3 2 2 1] and B[3 2 4 1]. Overall scale to double the size of line AB followed by two point perspective projection with COP along X-axis and Y-axis as Xc=10 and Yc=20 respectively. Also, write the corresponding vanishing points.
So, first of all let us double the size of the line segment. We have been given homogenous coordinates of A and B. Let us apply our scaling matrix to it.
![](https://static.wixstatic.com/media/738fdb_8aa789c93c264ecba80da002c3e1d0a3~mv2.jpeg/v1/fill/w_147,h_76,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/738fdb_8aa789c93c264ecba80da002c3e1d0a3~mv2.jpeg)
So the new coordinates are [6 4 4 1] and [6 4 8 1]. Now we can apply two point perspective projection along it. Please note that since it is happening along x axis and y axis we will take z=0 plane for this projection. The transformation matrix for two point perspective projection along z=0 plane is given by:
![](https://static.wixstatic.com/media/738fdb_6161492e3be54342a2e3eedd1c6fc695~mv2.jpeg/v1/fill/w_147,h_171,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/738fdb_6161492e3be54342a2e3eedd1c6fc695~mv2.jpeg)
Now let us apply transformation matrix on our endpoints of the line.
![](https://static.wixstatic.com/media/738fdb_bf1a63243bfa4438975342f676e5a216~mv2.jpeg/v1/fill/w_147,h_76,al_c,q_80,usm_0.66_1.00_0.01,blur_2,enc_auto/738fdb_bf1a63243bfa4438975342f676e5a216~mv2.jpeg)
As we can see that although it was two point projection it turned out to be one point since both the vanishing points lie at the same coordinates.