3. Compute the touching point

(tapping and cymbal beats) - Great. So hopefully you're closer to developing a hypothesis for the relationship between the lengths of the segments in the diagram that describe exactly where the touching point is. Now, recall that we're interested in the formula for a touching point because that'll let us write a computer program to compute just points on the parabolic arc and not have to worry about where the string art lines are or even where the control points are. Let me tell you a little bit about how I came up with my hypothesis. So when we look at the diagram here with t set to the midpoint, that is, we're talking about a string art line that's halfway along the way in the construction. That is, this point is the midpoint of this segment. This point's the midpoint of this segment, and finally, the touching point looks like it's the midpoint of this string art line. So, in this case, all those ratios are the same. Let's go back to, say, t equals around a quarter, so this point is a quarter along the way here, this point is a quarter along the way here, and in this case, it looks like the touching point is also a quarter along the way of the string art line. So again, all the ratios are equal. Let's try one other value of t, say, something around 0.7, so this point is 0.7 along the way, this point is 0.7 along the way, and again, it looks like the touching point is about that same ratio, about 0.7 along the way. That leads us to guess that if this point is a fraction t along this line segment, and this point is a fraction t along this line segment that the touching point that we're after is that same fraction along this string art line. Now to turn this into formulas, let's start by labeling our points. Let's call the control points A B and C. This point here on AB, I'll point Q. This point here on BC, I'll call point R. We'll call the touching point that we're after point P. So the geometry says that if this is a fraction t along the way, that this in ratio t to one minus t. This is in ratio t to one minus t, and this is in ratio t to one minus t. The algebra that goes along with this geometry is that Q can be written as one minus t times A plus t times B because it's on the line segment AB. The point R is on the line segment BC, a fraction t along the way, so it can be written as one minus t times B, plus t times C. If P is a fraction t along the line segment QR, then it can be written again as one minus t times Q plus t times R. So with these three formulas taken together, we can compute any point on the parabola we like just by varying the value of t. The next exercise will give you some practice using these formulas to solve a few problems.