Are there any mathematical equations or a math procedure by which you can define a perspective drawing? I have seen so many mathematical procedures including matrix. But all don't seem to plot a perspective view. Why is it hard to establish simple vector equations by which you can input the true x,y,z coordinates of any point in the object and output X,Y coordinates of the corresponding point in perspective projection? Keep in mind that the perspective view is such as a photo taken by a camera. Hence, the location of the camera with respect to the object as well as the tilt angle of the camera (as per bird's and worm's eye views) can give different perspective views. Does anyone have any mathematical approach to execute such a process?
Coordinates of the plan
a(-0.38, 1.86).      b(0.18, 1.65)
c(0.38, 2.21).        d(-0.18, 2.41)
The height of the cube is   (0.61)
Please, show me how you get the coordinates of eight points of the perspective drawing. Thanks!


 A: Imagine a central projection, with $(0,0,0)$ as the center and $z=1$ as the image plane. Then you draw a 3D point $(x,y,z)$ at position $(x/z,y/z)$. That's the core idea behind homogeneous coordinates: any point in the direction $(x,y,1)$, i.e. any multiple thereof, represents the point $(x,y)$.
If your camera or your image plane is somewhere else, you'd usually first apply a transformation to reduce that to the situation just described. If you want to include translations in 3d space, you would input homogeneous 3d points, i.e. vectors with four elements, so the whole projection would be a $3\times4$ matrix converting homogeneous input to homogeneous output coordinates.
A: 
Perspective equations
I am the founder of these equations. They were the source code of a perspective machine patented in 1981 in England (Patent No. GB-2069938A). I was 19 years old when I invented it. These equations are the only simple vector equations that define the perspective projections of any object rotating in any direction. The equations also involve the zoom factor (F) and the tilt angle of the camera (n°) to plot bird's-eye and worm's-eye views. You only have to input x,y,z coordinates of any object to get the perspective projection coordinates (X and Y). The equations are easy to prove, but the diagram is based on three point perspective procedure which is hard to understand. There isn't a lot of resources to clarify three point perspective except in my book. I put them publicly for sake of the community in large if they serve the purpose for descriptive geometry for 3D-CAD software development. I will be pleased to answer any of your questions.
Here you can find the complete research and the proof of the above equations:
https://www.quora.com/What-is-the-simplest-mathematics-to-generate-3D-models/answer/Dhirgham-Murran-1
I also put on Google Drive the Excel file, by which you can input (x,y,z) coordinates together with the tilt angle of the camera (n) and the magnification constant (i.e. zoom in/out "F") and the output perspective coordinates (X,Y) will be automatically given based on the equations.
https://drive.google.com/file/d/14fczx3h4TosEAYVCuJ9reCpEIqPGb_9y/view?usp=drivesdk
