Question

I have a line from (a, b) to (x, y), and I would like to draw a line starting at (x, y), with length ℓ, that makes an angle of θ with the original line.

How do I compute the coordinates of the endpoint of this new line? See the diagram:

Answer 1

It’s nearly always simpler to use vector algebra for this kind of thing, rather than Cartesian coordinates. Let’s start by labelling the points:

Let R(θ) be the matrix that rotates by θ radians counter-clockwise:

Then compute:

v = B − A (the vector from A to B)

v̂ = v / |v| (the unit vector in the direction of v)

ŵ = R(−θ) v̂ (the unit vector in the direction of BC; your rotation is clockwise, so we need R(−θ) here, not R(θ))

w = ℓ ŵ (the vector of length ℓ in the direction of BC)

C = B + w

This approach avoids the need to compute an arctangent, which would need some care (if done naïvely, it runs into trouble when B is vertically above or below A; but most languages have a function like atan2 for handling this case).

In any sensible programming language with a vector library you should be able to write this as a one-liner, perhaps like this:

C = B + (B - A).unit().rotate(-theta) * l