Linked Questions

0
votes
0answers
15 views

Constant speed/distance on bezier curve with any number of points [duplicate]

I'm working on a graphics project and I'd need to find a way to get a constant speed/distance for a bezier curve when lerping with T (between 0 and 1). I've found a few answers, but the problem is ...
30
votes
8answers
28k views

Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated (...
6
votes
4answers
1k views

Difficult Integral: $\int\frac{x^n}{\sqrt{1+x^2}}dx$

How to calculate this difficult integral: $\int\frac{x^2}{\sqrt{1+x^2}}dx$? The answer is $\frac{x}{2}\sqrt{x^2\pm{a^2}}\mp\frac{a^2}{2}\log(x+\sqrt{x^2\pm{a^2}})$. And how about $\int\frac{x^3}{\...
3
votes
4answers
2k views

Using Runge-Kutta-Fehlberg 4-5 for higher dimension systems

When applying RKF45 algorithm to a first order ODE with higher dimensions, e.g. $f(t,y_1,y_2)$ and $f(t,y_1,y_2,y_3)$, is the procedure simply a matter of applying RKF45 to each dimension in turn? I.e....
1
vote
2answers
933 views

Finding a quadratic Bézier curve of length $l$ between two points

I have two points $P_1$ and $P_2$ in the plane. For each of the points, I have two vectors $v_1$ and $v_2$. I want to find a quadratic Bézier curve from $P_1$ to $P_2$ of length $l$ leaving $P_1$ in ...
3
votes
1answer
751 views

Find the arc-distance between the start and end points of a Bézier curve

Given that I have the start, end and control points for a linear Bézier curve, I am trying to find the arc-distance between the start and end points. Google seems to be failing me this morning; can ...
0
votes
0answers
907 views

Calculating the length of a NURBS curve

I'm attempting to find the length of a NURBS curve, but I'm not having any luck (I'm also not entirely sure if NURBS are more of a programming thing, I initially asked this question over on Stack ...
1
vote
1answer
466 views

Resample Bézier Curve with curvature and number of points constraints

I have an algorithm that implements an uniform resample process throughout a Bézier curve. This is done using a chord parametrization process. However, the results achieved do not accomplish my needs....
2
votes
3answers
71 views

What should be minimized in order to best approximate a curve?

I have a smooth parametrized curve $r(t), t \in [0,1]$ in the plane and I want to find a sequence $r(t_1), r(t_2), ... r(t_n)$ which "best" approximates the curve. I'd like to know how to define "best"...
3
votes
1answer
112 views

What method are there for “numerically” computing arclengths!

I know the originals formula for arc-length is: $$\int_{a}^b \sqrt{1+{f'(x)}^2}$$ However most of the formulas don't have closed formed solutions, and are unsolvable in terms of this equation. So ...
0
votes
4answers
171 views

Length of a curve without function?

I need to find the length of this curve: I don't have a function but I do have 3 sets of coordinates: $(0, 51)$, $(337, 674)$, and $(1022, 1022)$ The $(337, 674)$ set refers to the tangent and it ...
2
votes
1answer
64 views

How did length of base polyline relate with length of Bézier curve?

Let cubic Bézier curve $C$ be based on points $p_0, p_1, p_2, p_3$. Suppose $L$ is the polyline through $p_0, p_1, p_2, p_3$. Is there some well known analytical relation between lengths of these two ...