Note on Projective Coordinates

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After the perspective projection matrix is applied, each vertex undergoes "perspective division.

Explaining Homogeneous Coordinates & Projective Geometry — Tom Dalling

Continuing with the example above, the perspective division step would look like this:. In GLM, this perspective projection matrix can be created using the glm::perspective or glm::frustum functions.

In OpenGL, perspective division happens automatically after the vertex shader runs on each vertex. One property of homogeneous coordinates is that they allow you to have points at infinity infinite length vectors , which is not possible with 3D coordinates. What use does this have? Well, directional lights can be though of as point lights that are infinitely far away. When a point light is infinitely far away the rays of light become parallel, and all of the light travels in a single direction. This is basically the definition of a directional light. This is more of a traditional convention, rather than a useful way to write lighting code.

Directional lights and point lights are usually implemented with separate code, because they behave differently. A typical lighting shader might look like this:. Matrices for translation and perspective projection transformations can only be applied to homogeneous coordinates, which is why they are so common in 3D computer graphics.

Toggle navigation Tom Dalling. Home Blog. The Math Now, let's look at some actual numbers, to see how the math works. Depth of field, which will be described and implemented at the end of this section, simulates the blurriness of out-of-focus objects that occurs in real lens systems. ProjectiveCamera implementations pass the projective transformation up to the base class constructor shown here.

Set of Points on Elliptic Curve in Projective Coordinates

This transformation gives the camera-to-screen projection; from that, the constructor can easily compute the other transformation that will be needed, to go all the way from raster space to camera space. The only nontrivial transformation to compute in the constructor is the screen-to-raster projection. Finally, we scale by the raster resolution, so that we end up covering the entire raster range from left-parenthesis 0 comma 0 right-parenthesis up to the overall raster resolution.

An important detail here is that the y coordinate is inverted by this transformation; this is necessary because increasing y values move up the image in screen coordinates but down in raster coordinates. The orthographic transformation takes a rectangular region of the scene and projects it onto the front face of the box that defines the region. The orthographic camera constructor generates the orthographic transformation matrix with the Orthographic function, which will be defined shortly.


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To do this, the scene is first translated along the z axis so that the near plane is aligned with z equals 0. Then, the scene is scaled in z so that the far plane maps to z equals 1. The composition of these two transformations gives the overall transformation.

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The directions of the differential rays will be the same as the main ray as they are for all rays generated by an orthographic camera , and the difference in origins will be the same for all rays. Therefore, the constructor here precomputes how much the ray origins shift in camera space coordinates due to a single pixel shift in the x and y directions on the film plane.

We can now go through the code to take a sample point in raster space and turn it into a camera ray. First, the raster space sample position is transformed into a point in camera space, giving a point located on the near plane, which is the origin of the camera ray. Because the camera space viewing direction points down the z axis, the camera space ray direction is left-parenthesis 0 comma 0 comma 1 right-parenthesis.

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Depth of field will be explained later in this section. Finally, the ray is transformed into world space before being returned. The implementation of GenerateRayDifferential performs the same computation to generate the main camera ray.

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The differential ray origins are found using the offsets computed in the OrthographicCamera constructor, and then the full ray differential is transformed to world space. The perspective projection is similar to the orthographic projection in that it projects a volume of space onto a 2D film plane.

However, it includes the effect of foreshortening: objects that are far away are projected to be smaller than objects of the same size that are closer. The perspective projection is a reasonably close match to how an eye or camera lens generates images of the 3D world.

How to Apply Affine Transformations

The perspective projection describes perspective viewing of the scene. Points in the scene are projected onto a viewing plane perpendicular to the z axis. See more details on OpenGL Transformation.

Therefore, we can set the w-component of the clip coordinates as -z e. And we set w c to -z e earlier, and the terms inside parentheses become x c and y c of the clip coordiantes. Finding z n is a little different from others because z e in eye space is always projected to -n on the near plane. But we need unique z value for the clipping and depth test. Plus, we should be able to unproject inverse transform it. Since we know z does not depend on x or y value, we borrow w-component to find the relationship between z n and z e.

In eye space, w e equals to 1. Therefore, the equation becomes;.

Projective Plane Rotating in 4D

To find the coefficients, A and B , we use the z e , z n relation; -n, -1 and -f, 1 , and put them into the above equation. To solve the equations for A and B , rewrite eq. Substitute eq. Put A into eq. We found A and B. Therefore, the relation between z e and z n becomes;.

This projection matrix is for a general frustum. If the viewing volume is symmetric, which is and , then it can be simplified as;.