Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
There are some situations that are useful to learn to parameterize. We commonly parameterize line segments, and require knowledge of the starting and ending positions. In order to understand how to parameterize line segments, students should understand the concept of parametric equations. The tools we use to parameterize a line can be useful when understanding how to parameterize a circle.
I want to talk about how to get a parametric equation for a line segment. Now let's start with a line segment that goes from point a to x1, y1 to point b x2, y2. And now we're going to use a vector method to come up with these parametric equations. First of all let's notice that ap and ab are both vectors that are parallel. So ap equals t times ab some scalar t times ab and you know that because ap is smaller than ab that this is going to be some kind of compression that means that t is going to be between 0 and 1 as long as point p is between these 2 guys. Now let's take a look at the components for ap and ab I've named point p x and y I'm going to use this point to derive my equation, my equation has to have x and y.
Let's say that these are numbers that I know these coordinates here and here, ap has components x-x1 and y-y1 and ab has components x2-x1 and y2-y1. So I have a scalar of multiplication here and I'm going to multiply the t through and I get tx2w-tx1, ty2-ty1 and that equals x-x1, y-y1, now to go from a vector equation to parametric equations all I have to do is separate these by components. So x-x1=tx2-tx1, tx2-tx1 let me add the x1 to both sides and I get x equals and I'm going to pull this guy in front, x1-tx1 and then plus tx2 and that gives me x equals, notice that I have a common factor of x1 I can pull out the 1-t and I'm left with x1, 1-t times x1+t times x2.
Now because t is between 0 and 1 this is like a weighted average of x1 and x2 and now the same thing is going to happen for y, y-y1 equals ty sub 2 minus ty sub 1 again I'll move the y1 term in front and I'll add y1 to both sides I'll get y1-ty1+ty2 so this is after I factor the y1 out, I get 1-t, y1+ty2=y. So these are my 2 equations right here and then putting them together x=1-tx1+tx2, y=1-ty1+ty2 and these equations work for t again between 0 and 1. So all you need to come up with a parametric equations for a line segment are the coordinates of the 2 end points x1 y1 and x2 y2 and you can always use this parameterization to get you that line segment.