If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:3:25

thanks for hanging in there I know this discussion is getting a little bit technical but we finally have all the tools we need to complete the derivation of a formula for the touching point on a parabola but before we continue let's back up just a little bit and remind ourselves why we're doing this again well we need that touching point formula so that shots like this in brave can be computed really efficiently right because that touching point will allow us to write computer programs to draw each blade of grass without having to draw all the individual string art lines so to turn this into formulas let's again label things so this light magenta line is controlled by the parameter T so I'm going to label as before this point Q and this point R the dark magenta line is controlled by the parameter s so I'm going to call this point Q Prime and this point R prime now let's start writing down a few things we know well we know that Q is a fraction T along the line segment a-b which means I can write Q is 1 minus T times a plus T times B similarly R is a fraction T along the line segment BC so it can be written as 1 minus T times B plus T times C similarly Q prime is s along the way from A to B so I can write Q prime as 1 minus s times a plus s times B and I can write R prime finally as 1 minus s times B plus s times C ok now this intersection point here that we're looking for P is somewhere on the line segment QR but we're on the line segment is it well I'm going to prove in a second that it's at fraction s that is I claim that P can be written as 1 minus s times Q plus s times R now if that's true something nice happens because as s approaches T this expression here approaches 1 minus T times Q plus T times R and that's the thing I ultimately want to prove so the only thing left to show is that the intersection can be written this way why should that be the case what I'm going to do is I'm going to substitute this expression for Q and here this expression for R in here and if I do that and rearrange I'll leave that rearrangement to you but the result is that P can be written as 1 minus s times 1 minus T times a plus s times 1 minus T plus T times 1 minus s times B plus s T times C and now if I rewrite this using these expressions for Q prime R prime I see I can write P as 1 minus T times Q prime plus T times R Prime well this expression says that P is somewhere on the line segment Q prime R prime and this expression says that P is somewhere on the line segment QR and the only point that can be on both line segments is the intersection point so our proof is complete bullseye