# Find angle at point on bezier curve

I have two end points and two control points. I am using these points and this link. i have found a point on bezier curve. Now i would like to find angle at this point on bezier curve. Is there any formula?

• What do you mean by an angle of a curve? An angle is formed by the intersection of two (differentiable) curves. What is you angle relative to? Aug 27, 2013 at 17:26

A cubic Bézier curve has an equation of the form $$\mathbf P(t) = (1-t)^3\mathbf P_0 + 3t(1-t)^2\mathbf P_1 +3t^2(1-t)\mathbf P_2 + t^3\mathbf P_3$$ When you say you want the "angle" of the curve, I suppose you mean the angle between the curve's tangent and the $$x$$-axis. If this is what you want, then here's how to get it:
If you differentiate the curve equation, you'll get $$\mathbf P'(t) = (1-t)^2(\mathbf P_1 - \mathbf P_0) + 2t(1-t)(\mathbf P_2 - \mathbf P_1) + t^2(\mathbf P_3 - \mathbf P_2)$$ As you probably know, $$\mathbf P'(t)$$ is a vector that's in the direction of the tangent line of the curve at parameter value $$t$$. So, you just need to find the angle between this vector and the $$x$$-axis. If the vector is $$(u,v)$$, then the angle is $$\arctan(v/u)$$. If you're writing code, compute $$\text{atan2}(v,u)$$.