(Geometry: intersection) Suppose two line segments intersect. The two endpoints for the first li

step 1 Okay, good day, ladies and gentlemen. So today we're looking at problem number 7 .9, and it's on

(Geometry: intersection) Suppose two line segments...

Key Concepts

System of Linear Equations

When two lines are represented by their equations, finding their intersection point involves solving a system of linear equations. This approach uses methods like substitution or elimination to find the unique solution (if it exists) that satisfies both equations simultaneously, indicating the intersection point of the lines.

Line Equation

A line equation describes an infinite set of points on a line using algebraic expressions, typically in slope-intercept or point-slope form. By deriving the equation from two endpoints, one can analyze the line's behavior and characteristics, such as its slope and intercept, which are critical in determining points of intersection.

Two-Dimensional Coordinate Geometry

This concept involves representing points in a plane using an ordered pair of numbers (x, y). It is fundamental for describing the location of endpoints of line segments and for visualizing geometric objects and their relationships in a coordinate system.

Line Segment

A line segment is a part of a line defined by two distinct endpoints. In computational geometry, line segments are used to represent finite portions of lines, and understanding their properties is essential for determining relationships such as intersections between segments.

Transcript

00:03 Okay, good day, ladies and gentlemen.

00:07 So today we're looking at problem number 7 .9, and it's on intersection.

00:13 Now, this problem is a little bit different than some of the ones we've been doing recently, or that i've done in the last few videos.

00:22 Basically, it's sort of asking us to apply one of the classes we've used earlier, and before we get to the actual implementation of it, i want to sort of sketch out what's going on and then we'll get to the next step.

00:46 So if you read the problem, you understand basically what we have.

00:54 So we have a line, so we have these points here.

01:00 We'll say this point is, we'll call this x1, y, one.

01:08 Okay.

01:10 And we have another point over here.

01:15 We'll call this x2, y2.

01:29 And we have a line between these two, so a line like this, which, of course, goes on.

01:36 Like that, added finitum, you know, like that.

01:42 Okay.

01:44 And a couple equations we can get for this line is we have to use slope.

01:52 The slope, the slope, the slope, the slope formula to get the equation for the line of the form x, minus x1 uh sorry yeah times y um actually i want to write it like this i want to write it uh y 1 minus y 2 um times x plus um uh x 2 minus x1 y 2 equals x 2 minus x 1 y to equal to this is let me see it's supposed to be x2 minus x1 times some constant we'll call that just e okay and the the way i got this is that you use the so the slope intersect force i mean there's ways you can write this, but one way to do it is to say, well, y is equal to mx plus b, okay? and we have that m is equal to y1 or y2 minus y1.

03:38 And this is sort of the rise overrun idea, which is x2 minus x1.

03:46 Okay.

03:47 And then when i multiply through, i end up with this equation over here once i subtract.

03:55 The main reason i want this form, okay, is because when i do what i want, when i, when i'm using the linear equations.

04:11 If you recall our linear equations module were at, oops, so our linear equation module had the form like this.

04:22 It had ax plus b, y equals to e, and cx plus by equals to f.

04:36 Okay? and the module then, of course, we plugged in in the a, b, c, d, e, and f, and spit out values for the solution.

04:47 So what i wanted to do, basically, is get this into something like this form.

04:55 So i'd have, you know, the first equation, my a now would be y1 minus y2.

05:01 My b would be x2 minus x1.

05:04 And my e would be, well, in this case, it would actually be the intersect value, and the e here actually is, if you solve, you get that e is actually equal to, this should be, let's see, y1, i should be careful about how i do this.

05:32 It's y1 minus y2 times x1 plus what is that? yeah, x2 minus x1 times y1.

05:57 And actually, now they look at this, you could have simplified that.

06:01 I'm just realizing now that could have simplified more than i realized.

06:05 But the key is that this would be the e value that you use.

06:14 But you put all these values in from the variables you get.

06:23 When you start the program, you gather x1, y, 1, x2, y3, and x4, y4, which, of course, would be something like this.

06:40 We'd have like the points x3, y 3, and then some point over here.

06:49 Whoops, sorry, a different thing.

06:59 Irritating as heck, i can tell you that.

07:02 Irritating as heck.

07:04 X for y, four.

07:07 And again, we're going to get a line three.

07:12 These like this, so on and so forth.

07:16 And then you're going to get this line, and this line will go here, the top line goes there, and then you'll end up with the intersection right there.

07:27 And, of course, it's not always going to happen.

07:29 You're going to have any intersection in particular.

07:34 If the lines are parallel to each other, there will be no intersection.

07:38 And that's really the one case that you'd worry about if they're parallel...

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