Given a 2D cubic Bezier segment defined by $P_0, P_1, P_2, P_3$, here’s what I want: A function that takes the segment and outputs the maximum curvature without using an iterative approach. I have a function that finds the maximum curvature at the moment, but does this using Brent’s Method to search a range of […]

I need to write an OpenGL program to generate and display a piecewise quadratic Bézier curve that interpolates each set of data points: $$(0.1, 0), (0, 0), (0, 5), (0.25, 5), (0.25, 0), (5, 0), (5, 5), (10, 5), (10, 0), (9.5, 0)$$ The curve should have continuous tangent directions, the tangent direction at each […]

For a planar cubic Bezier curve $B (x(t),y(t))$, I would like to find the values of parameter $t$ where the curvature (or curvature radius) is greatest/smallest. The formula for curvature is: $$r = \dfrac{(x’^2+y’^2)^{(3/2)}}{x’ (t) y”(t) – y'(t) x”(t)}$$ The problem is that there is that square root in it so I was wondering whether […]

If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?

Recently I read some course notes and articles on Bézier curves. They all sum up the properties of Bézier curves, like the partition-of-unity property of the basis functions (Bernstein polynomials), variation diminishing property of the curve (the curve doesn’t wiggle/oscillate more than the control polygon does), and also the convex hull property. Apparently the latter […]

I have to find points (say 10 points) in Bezier curve with 2 control points such that they are at equidistant positions in the curve. Currently I am using the following formula which gives me points but they are not equidistant. t = ………..from (1/10 to 10/10); q1 = t * t * t * […]

I’m trying to figure out a generic way of determining a series of points on a Bézier curve where all the points are the same distance from their neighboring points. By distance I mean direct distance between the points not distance along the curve. See the image below I’ve written a program that will solve […]

We can draw an infinite number of parabolas that pass through three given points $A$, $B$, $C$ (in that order). For each such parabola, we take the tangent lines at $A$ and $C$, and intersect them to get a point, $P$, which is called the “apex” of the parabola segment, in my business. What is […]

When someone has the 4 control points P0, P1, P2, P3 of a 2D cubic Bézier curve, that person can calculate a series of hundreds of points along the curve that start from P0 at t=0 and end at P3 at t=1 (but, in general, those points never hit P1 or P2). If that person […]

I’m working on an engineering project, and I’d like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple – a gentle curve which begins and ends horizontal. Is there a simple equation for this curve? Or perhaps two equations, one for each half? […]

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