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I have been studying computer graphics, from the book Fundamentals of Computer Graphic (but the third edition), and I lastly read about projections. Though, I didn't exactly understand what's the difference between orthographic and perspective projection? Why do we need both of them, where are they used? I also would like to learn what is perspective transform that is applied before orthographic projection in perspective projection. Lastly, why do we need the viewport transformation? I mean we use view transformation if the camera/viewer is not looking at $-z$-direction, but what about the viewport?

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Orthographic projections are parallel projections. Each line that is originally parallel will be parallel after this transformation. The orthographic projection can be represented by a affine transformation.

In contrast a perspective projection is not a parallel projection and originally parallel lines will no longer be parallel after this operation. Thus perspective projection can not be done by a affine transform.

Why would you need orthographic projections? It is useful for several artistic and technical reasons. Orthographic projections are used in CAD drawings and other technical documentations. One of the primary reasons is to verify that your part actually fits in the space that has been reserved for it on a floor plan for example. Orthographic projections are often chosen so that the dimensions are easy to measure. In many cases this is just a convenient way to represent a problem in a different basis so that coordinates are easier to figure out.

ortho

Image 1: A number of useful orthographic projections for same object (and projection rule). The last on on the right is a special case called isometric having the property that cardinal axe directions are all in same scale.

A perspective projection is needed to be able to do 2 and 3 point perspectives, which is how we experience the world. A specific perspective projection can be decomposed as being a combination of a orthographic projection and a perspective divide.

enter image description here

Image 2: 2 point perspective note how the lines in prespective direction no longer are parallel

Viewport transformation allows you to pan/rotate/scale the resulting projection. Maybe because you want a off center projection like in cameras with film offset, or you have a anisotropic medium for example. It can also be convenient for the end user to zoom into image without changing the perspective in the process.

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