line equation ? and intersection ?

Question

A line segment is given by its endpoints (10,10) and (32,15)

a) what is the point-slope formula for this line ?

b) what is the parametric equation for this line ?

c) Suppose we have a window with lower left corner (8,12), width of 20 pixel and height of 20 pixels. Find the intersection of this line with the edges of the window. And specify the new clipped vertices.

Answer 1

a) Slope = Δy/Δx = (y₁-y₀)/(x₁-x₀) = (15-10)/(32-10) = 5/22

Point-slope form: y-y₀ = m(x-x₀), which in this case is
y-10 = 5/22 (x-10)

b) There are many ways of parameterizing this line. One way is to use x=x₀+Δxt and y=y₀+Δyt, which has the advantage of giving you the point (x₀, y₀) for t=0 and (x₁, y₁) for t=1. Thus, in this case you would have:
x=10+22t
y=10+5t

c) The line has positive slope, and the lower-left corner lies above the line, which means that the line intersects the bottom edge of the window (rather than the left edge), so the first point of intersection is at y=12, which means:

12-10 = 5/22 (x-10)
2=5/22 (x-10)
44/5 = x-10
x=94/5

Similarly, the upper-right corner (at coordinates (28, 32)) lies well above the line, so the second point of intersection is with the right wall at x=28, and y=5/22 (28-10) + 10 = 45/11 + 10 = 155/11.

Therefore, the new clipped vertices are (94/5, 12) and (28, 155/11).