Plane Curve with Constant Angle

Question

Find a plane curve starting at (1,0) with a constant angle α = 75° between the tangent and position vector at each point.

Answer 1

This problem is solved more conveniently using polar coordinates.

Each point on the curve is given by a function r(θ).

This curve has the property that the angle ψ formed between the vector r and the tangent satisfies the identity.

tan(ψ) = r / (dr/dθ)

We are supposed to find the function r(θ) satisfying

tan(75°) = r / (dr/dθ)

This equation can be solved by separation of variables:

tan(75°)dr/r = dθ

tan(75°)ln(r) = θ + C (1)

The constant of integration, C, is determined from initial conditions:

The point (1, 0) is characterized by

r(0) = 1.

Plugging (r=1, θ=0) into equation (1) yields C = 0.

Thus r(θ) = exp(θ/tan(75°)).

The shape is a spiral.