# How to visualize angle of projection in Oblique Parallel Projection?

We know that in Oblique Parallel Projection Point (x,y,z) is projected to position (x_p,y_p) on the view plane.Projector (oblique) from (x,y,z) to (x_p,y_p) makes an angle alpha. with the line (L) on the projection plane that joins (x_p,y_p) and (x,y). Line L is at an angle phi with the horizontal direction in the projection plane.See this image1:

And in Oblique Parallel Projection Angles, distances, and parallel lines in the plane are projected accurately.For example see below image2:

My question is where is the angle alpha in image2, I mean I see the angle phi on the image , so where is alpha in that image to understand better?I want to see 12 edges of original image projected to view plane with projector and angle alpha,phi levelling.

N. B:1 -- I am following Hearn and Baker book which screenshot like this.

N. B. -- I want to understand just intuition in easy way rather than details.

• Alpha cannot be depicted on Image2, because it goes out of the plane. Commented Dec 6, 2021 at 19:05
• I’m voting to close this question because it is a cross post of engineering.stackexchange.com/questions/48585/… Commented Dec 6, 2021 at 19:06
• @Theraot could you provide the only the image "12 edges of original image projected to view plane with projector and angle alpha,phi levelling." Commented Dec 6, 2021 at 19:10
• I'm convinced the best way to visualize this would be with 3D graphics. Which is entirely odd because we would be using a projection to understand another one. I'll try to find if somebody has done it, I really don't want to program it. Commented Dec 6, 2021 at 19:20
• @Theraot I just want only the image. Commented Dec 6, 2021 at 19:27

On OP request, I'm posting the images I produced as answer.

These were made with Sketchup. They show a cube a plane where the cube is projected in an oblique projection. The blue sectors represent the angle alpha, and green sectors represent the angle phi. Observe that the angle phi exists entirely in the projection plane, while alpha does not. Instead alpha is the angle between the projection plane and the segment that goes from a point to its projection. As a consequence it is impossible to depict alpha on the projection plane, since it goes out of it. Furthermore, depicting alpha requires knowing the relative position of the object and the projection plane.

These images have perspective. It is odd to me that I'm using a projection to visualize another. However, I believe this is the best way to gain an intuition of it (actually I believe being able to orbit at will is better, but sans that here are multiple views).

I believe these are Cabinet, unless I'm confusing things.

Animation:

• on 3rd image only phi angle but why alpha is absent? Commented Dec 6, 2021 at 21:17
• @Ponting Alpha is on a plane you are looking on edge. Well, almost, due to the perspective, you can slightly see it: i.sstatic.net/tuDQl.png - Notice I only marked 4 of the 8 alpha, the other 4 are reduced to a line - But if I were using an orthographic projection, they would all be reduced to a line. By the same logic phi is not visible on the 6th picture. Commented Dec 6, 2021 at 21:23
• one thing tell projection of cube you take front face of original image is perpendicular parallel projection on projection plane? Commented Dec 6, 2021 at 21:45
• @Ponting Yes, I did project the closer face to the plane perpendicularly to it. Now I'm thinking I shouldn't have done that. I'll re do it (edit: I see, I made a cavalier projection, which is considered a type of oblique projection, but what we want in this case). Commented Dec 6, 2021 at 22:07
• in Cabinet also front face is also perpendicular to projected image. Commented Dec 7, 2021 at 10:04