Pixar in a Box
- Start here!
- Weighted average of two points
- Weighted averages
- 2. Where is the touching point?
- Exploring the parabola construction
- 3. Compute the touching point
- Calculating the touching point
- 4. How can we prove this?
- Touching point
- Bonus: Completing the proof
Okay we know how to calculate the touching point, great! Next let's think about how we can prove this is true.
Want to join the conversation?
- I agree that as s->t the intersection point get closer to parabola, but I don't get why when s=t that point lies exactly on parabola. How can intersection of two identically line segments be single point. Am I missing something here?(3 votes)
- As s -> t (but not s=t) the point seems to get closer &closer to the parabola. So we 'assume' that if s is infinitely close to t the point almost lies on the parabola. Notice that in the video the s line doesn't coincide with the t line cause then it won't have a specific intersection point.
Hope this helps. :)(3 votes)
- Actually, I don't get the meaning of proving the formula is true. It means it is true to describe the movement of the touching point by this formula?(2 votes)
- Yes, the point of the proof is to show that if you have a line that starts at a point, Q, which is t along AB, and ends at a point, R, which is t along BC, then it touches the parabola at a point that is t along line QR.(5 votes)
- Are two lines that stay on top of each other considered to be intersect?(0 votes)
- Any two or more lines that never touch are parallel, and a line that is itself touches in every point, so it is not parallel. It is also not intersecting because that would mean only one intersection. However, they are called coinciding lines because they are the same.(2 votes)
(steps and bouncing) (switch clicks) - You should now have some experience in computing points on parabolas, using the formulas that we wrote down in the previous video. These formulas are based on the hypothesis that all of these ratios are in the same proportion, and that proportion is governed by t. But now we're gonna prove that these formulas are actually correct. So I'm gonna use this version of the interactive in the proof. You'll have a chance to experiment with this interactive in a minute. And as before, I've got a string art line that is controlled by a parameter t. As I wiggle t back and forth, the string art line wiggles back and forth. The method I'm gonna use to find where the touching point occurs seems a little bit sneaky at first, but it's the simplest method I know. And what I'm gonna do is introduce another string art line, this one controlled by a parameter s. And what I'm gonna do is I'm gonna write down an expression for this intersection point here, this green point. Now, why would I do that? Well, the reason is, watch what happens as s and t get closer and closer together. So as I make s closer and closer to t, watch what happens to this intersection point. OK, it moves, and it gets closer and closer to the parabola, and exactly when s equals t, that intersection point lies exactly on the parabola. So if I can write down a formula for where that intersection point is, I can compute exactly where the touching point occurs. This is probably a good time to pause and let you experiment with this interactive to get some feeling for what we're about to do algebraically.