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I've watched the Pixar In A Box, Maths of Rendering Tutorial on Khan Academy. I've started writing a raytracing program and have come to the point where I must solve for t (the parameter for the parametric ray through the camera and a point on the view plane).

In the tutorial, this is done by simply substituting the parametric formula for each x, y, and z into the equation of the plane that the triangle lies on. This leaves a single unknown (t).

However, in all the examples and exercises that are given by the tutorial the camera is located at [0,0,0]. As I'm sure you'll understand, this is in many cases unrealistic. Because the camera is at 0, they simplify as follows:

$$i=\frac{1}{t}C+tP$$

becomes simply:

$$tP$$

Which makes for an easy solve. My question is when the camera is not at [0,0,0] should I:

  1. Calculate the coordinates of the plane relative to the camera. (So the camera acts as if it is [0,0,0]
  2. Solve the equation as a quadratic. I'm still confused a bit about why this is quadratic in the first place. Could someone explain this? And which solution do I use?

Of the two options, which is more acceptable for use in this community? Why? Is one more computationally efficient?

Edit: I think I have found a solution as follows: The parametric equation (which is effectively just a weighted average) is represented like this in the tutorial:

$$x(t) = \frac{1}{t}C_{x}+tP_{x}$$

But I realize now that is the same as:

$$x(t)=C_{x}+t(P_{x}-C_{x})$$

Which is dramatically easier to solve.

So my final question is, which of the three is the most acceptable way to solve for t? I'm assuming the third one.

wychmaster
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PolymorphismPrince
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1 Answers1

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Wow I'm stupid, the whole problem was I thought it was 1/t and it is actually 1-t in parametric form. Hope this helps someone anyway.

PolymorphismPrince
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