2. Parametric form of a ray
About this video
With A = (3, 2), B = (4, -1) and P = (2, 1/2), the video writes line AB as y = -3x + 11 and ray CP as y = 1/4 x, then finds the intersection I by solving the two equations together. Because rays in 3D have no slope-intercept form, it replaces this with a parametric function R(t) = (1 - t)C + tP, a weighted average of C and P where t = 0 gives C, t = 1 gives P, and t > 1 gives points beyond P.
You'll be able to
- write the slope-intercept form of a line through two points.
- find an intersection point by solving two line equations together.
- use R(t) = (1 - t)C + tP to find points on a ray for different values of t.
Key words
- Slope intercept form
- Writing a line as y = mx + b, where m is the slope and b is where it crosses the y-axis.
- Parametric function
- A function that gives the points on a line or ray using a single changing number, the parameter t.
- Weighted average
- An average where each value is multiplied by a weight that says how much it counts.
- Parameter
- The input number t that picks out one point on the ray.
Transcript
- Recall that to figure out where a two-dimensional ray intersects a line segment, we started by first introducing a coordinate system. Once we have our coordinate system, we can write our line AB in slope intercept form. Since in this example A has coordinates three two, and B has coordinates four negative one, the slope intercept form of AB is y equals negative three x plus 11.
Similarly, if I pick P to have coordinates two, one half, then the slope intercept form of the ray CP is y equals one quarter x. The point I we're looking for is on both of these lines. So if I sub x and I sub y represent the coordinates of I, then I sub y equals negative three I sub x plus eleven, because I lies on AB.
And I sub y equals one quarter I sub x, because I lies on the ray CP. Solving these two equations for the two unknowns gives us the coordinates I sub x and I sub y. Using the slope intercept form of the ray works in two dimensions, and is fairly simple to understand, but there's a problem when we generalize it to three dimensions.
The problem is that in three dimensions, the ray doesn't have a slope intercept form. So we'll have to throw out this representation of our ray in preparation for three dimensional raytracing. To represent our ray CP, we'll use something called a parametric function.
What I'm about to write looks a little strange at first, but bear with me, these functions start to become familiar with practice. My ray will be represented by a new function, R of t, that is the weighted average of C and P, where t is the weight. In particular, I'm going to write R of t as one minus t times C, plus t times P.
Notice what happens when t equals zero. One minus t is just one. So R of zero is C. And when t equals one, R of one equals P. That's convenient, because I can relabel C as R of zero, and I can relabel P as R of one. R of one half would be exactly halfway between C and P.
Values of t greater than one name points on the ray off in the scene beyond P. Before we continue, get some experience using this kind of parametric function in the next exercise.
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