I’m trying to find the intersection between a line segment and a quadratic bezier triangle for my OpenCL real time raytracer. The main recomendations I’ve seen are to try subdivision, or tensor product bezier patches. I’ve read in a few places that when testing a line segment against a quadratic bezier triangle, that you just […]

Budan’s and Vincent’s theorems can be used to isolate the real roots of a real polynomial. I have read papers which compared it favorably to other root finding methods. However, roots can also be isolated by converting the polynomial into Bernstein basis and splitting the curve by the means of De Casteljau’s algorithm. Both the […]

What is the explicit formula for a unit tangent vector to a Bezier curve? I.e. if the formula for a Bezier curve is $\mathbf{B}(t) = \sum_{i=0}^n\binom{n}i(1-t)^{n-i}t^i\mathbf{P}_i$, what is its unit tangent?

I’ve got a 2D quadratic Bézier curve which, by construction, is a f(x) function : no loops, a single solution for each defined x. Is there a common mean to convert this curve to a 3rd degree polynomial ? 3 should be enough since there can only be two “bumps”. Thanks!

I’m developing an iPhone app that allows users to cut out part of an image from its background. In order to do this, they’ll connect a bunch of bezier curves together to form the clipping path. Rather than have them move the control points of the curves in order to change the curve, I’d like […]

The motivation for this is Bezier curves. But, if you don’t know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to construct a quadratic Bezier curve that passes through some given points. First, the standard approach: I would use three […]

Given two closed shapes made up of Bézier curves (and/or straight lines), I’m looking for an efficient way of calculating the resulting shape of the following Boolean operations: union difference intersection slice (imagine each of the shapes in the Venn diagram as its own shape; this operation is optional and can be expressed as a […]

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have the same x-axis coordinate My question is: How do you express the X-axis coordinate of the inflection point of […]

A quadratic Bézier curve is a segment of a parabola. If the $3$ control points and the quadratic Bézier curve are known, how do you calculate the equation of the parabola (which is an $y=f(x)$ function) that this Bézier curve is a part of? (algebraically) So in the image, the red curve is known, how […]

I have a situation where I have an arc that was mangled (irrelevant: by c#’s GraphicsPath.AddArc() function). The original arc is guaranteed to be circular, and the new data I have describes the Bézier approximation for the arc instead. I’m not hugely up on Béziers, or complex geometry, so am hoping someone can help me! […]

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