Questions on Bézier curves, curves that are frequently used in computer graphics.

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3
votes
2answers
874 views

How to calculate the square area under a Bezier curve?

I did search at Google and this website before asking this question, so sorry if this somehow has already been answered and I didn't notice. BTW I'm a humanities scholar and not a trained ...
2
votes
5answers
973 views

Rough y(x) approximation for simplified Cubic Bezier curve

I need to get a very rough (and fast) $y(x)$ approximation of a simplified Cubic Bezier curve to use in my animation code, where there's only one control variable: $$ P_0 = (0, 0)\\ P_1 = (0, 0)\\ ...
1
vote
1answer
635 views

Finding the mid-point of a B-spline curve.

As part of my computer graphics, we've learnt about Bezier and B-spline curves. I'm going over some questions in preparation for my exam and I've come across a past exam question. The question is "A ...
1
vote
0answers
143 views

Approximating Bezier curves

I would like to approximate one cubic Bezier curve with two quadratic ones. In other words, I would like to split a cubic curve at some parameter t and approximate ...
1
vote
1answer
2k views

Calculate intermediary control points in Cubic Bezier Curves

I need to programatically generate two-dimensional circles of various dimensions, knowing only their radius and position. The circles will be drawn by employing 4 cubic Bezier curves. How should I ...
3
votes
2answers
502 views

Bezier curvature

I'm trying to understand quadratic Bezier curves but I cannot get pass one thing. Please, what is a "curvature" and how can I calculate it? I'm asking because I found for instance: ...
1
vote
0answers
30 views

Is it possible to change a piece of curve's interpolation type of a B-Spline via modifying knots?

I am going to implement a curve editor based on (cubic) B-Spline. Sometime the user may change a piece of curve's interpolation type, that is, use linear/constant value between two consecutive ...
1
vote
1answer
281 views

An almost straight curve with infinite curvature?

I played around with computing the curvature of some curves, and found this weird example that is driving me nuts. Consider the following (Bézier) curve (on a plane, the first point is $[-1,0]$): ...
2
votes
2answers
356 views

Can an involute gear profile be modeled with a Bézier curve?

In the context of a game, I want to draw gears. The most common curves available on the platforms I'm using are third degree Bézier curves. Is there an exact representation of the involute gear ...
1
vote
0answers
74 views

How can the equation of a Bézier curve be transformed from a Bézier basis function to a bivariate function?

Several nights ago, I was researching the problem of identifying self-intersections in arbitrary curves, particularly Bézier curves. (The reason being is that I want to write a program that inserts ...
7
votes
1answer
2k views

Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]

Update Alternative Statement of Problem, with New Picture Given three points $P_1$, $P_2$, and $P_3$ in the Cartesian plane, I would like to find the ellipse which passes through all three points, ...
1
vote
1answer
2k views

Cubic Bezier Curves - Calculate Y for any given X [duplicate]

Possible Duplicate: Is there an explicit form for cubic Bézier curves? I want to calculate Y for any given X of a bezier to help me chart a graph. X represents time and Y represents ...
2
votes
3answers
409 views

How can I find out 2 unknowns in a cubic equation?

I need to give a bit of a background first, so please bare with me. I have a set of values that represent servo motor position values. By default I end up with a large set of values and I'd like to ...
0
votes
1answer
541 views

Function to represent curve between two 2D points

I need a (simplest) function that interpolates values in range from predefined point $A$ to $B$ with rules: it must be smooth curve direction near $B$ must be the same as predefined $D$ vector ...
0
votes
1answer
321 views

Reconstruct Control points in a Bézier Curve?

I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
4
votes
1answer
1k views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
1
vote
0answers
102 views

How can I calculate all possible Bézier handle points in order to make the curve to a given length?

Given two anchor points and a handle point of a cubic Bézier curve, how can I calculate the other handle point in order to make the curve length to a fixed value? What kind of orbit will it be? ...
1
vote
1answer
573 views

Finding point of inflection on a Bézier Curve

I need to determine the first point of inflection on a Bézier curve, if it exists, for a computer graphics application. My original idea was to iteratively walk the curve, evaluating 2nd derivatives ...
2
votes
1answer
220 views

Motion on a parametric surface

Please excuse what will surely turn into a long rambling question, full of incorrect terminology. I'm trying to figure out the mathematics of moving on a parametric surface - that is, for some ...
3
votes
1answer
848 views

Finding Y given X on a Cubic Bezier Curve?

I just asked this in the Computing sections but they sent me here: "So I've been looking around for some sort of method to allow me to find the Y-coordinate on a Cubic Bezier Curve, given an ...
0
votes
1answer
202 views

How can a Bézier curve be periodic?

As I know it, a periodic function is a function that repeats its values in regular intervals or period. However Bézier curves can also be periodic which means closed as opposed to non-periodic which ...
5
votes
1answer
4k views

How to derive the equation for a bézier curve

So, I remember a while back there was a maths competition and we were given a curve that we needed to write an equation for. I just skipped the question since I didn't even know where to begin. I ...
0
votes
1answer
694 views

A function that can smooth out line joining three points

I want to know if there is a function with which we can smooth out line joining two( or more )points . I've read we can do it with Quad and cubic splines or something but I am not clear with them. i ...
1
vote
2answers
448 views

Shifting a quadratic Bézier curve

I have a quadratic Bézier curve. I want to make a path that runs along that curve, whose width is 'w'. (So at any point in time along the exact center of that path, you will be also on the original ...
1
vote
1answer
890 views

How to merge two poly Bézier curves?

I am battling with a programming problem. Given two or more overlapping and interacting Bézier polygons, how can I perform merge (union) operations on a list of input Bézier polygons so as to produce ...
0
votes
1answer
804 views

Newton's Method, and approximating parameters for Bézier curves.

I've been wanting, for quite a while now, to polish up some source code I wrote for approximating arbitrary Bézier curves to given series of points. I managed to accomplish quite a bit, but I hit a ...
0
votes
1answer
200 views

How to fly a curve from one heading to another using only roll and pitch.

I have 3 perpendicular vectors representing an object in 3d space... Heading Right Up ...and I would like to be able to 'fly' this object so that it ends up at a ...
1
vote
1answer
288 views

Detect “Kinks” in Parallel Lines to Bezier Curves [x-post]

I feel like this is just as much a mathematics question as it is a programming question so I figured it couldn't hurt to cross post my question here. Original Post: ...
5
votes
4answers
3k views

Why is the Convex Hull property (e..g of Bézier curves) so important?

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), ...
2
votes
3answers
2k views

Can a rational Bézier curve take exactly the same shape as a part of the sine function?

I'm wondering whether a rational Bézier curve could take exactly the same shape as a part of the sine function. The best way to check this seems like this: Find a part of the sine function such that ...
1
vote
0answers
84 views

How to approximate a trigonometric curve by Bezier curves?

Let me ask how to approximate a trigonometric curve by Bezier curves? Is there any known algorithm? Thank you in advance.
1
vote
1answer
302 views

How to draw a smooth curve between 2 points given the 2 tangents at them?

Let me ask a question , given 2 points on the XY plane and given the 2 tangents at them, how to compute an arbitrary chosen smooth curve passing the 2 given points. For details, traveling along the ...
2
votes
1answer
221 views

Cubic curve software that generates equation

Is there a free / open-source program that would allow me to drag and drop points of a cubic curve and displays the equation? There are a number of equations that I could use in my game project and ...
10
votes
1answer
8k views

What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?

I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and ...
1
vote
1answer
423 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 ...
4
votes
2answers
294 views

Get $t$ of ascending Bézier curve from $x$

I have an ascending cubic Bézier curve. ($x_0 \leq x_1 \leq x_2 \leq x_3$) Considering this property, there is always one and only one $y$ value per $x$ value. The point ($x, y$) along the curve is ...
0
votes
1answer
627 views

Point projection on curve

Point projection on Bézier curves can be easily accomplished using Newton Iteration to try to minimize the dot product between the vector connecting the point P and its projection on curve C and the ...
1
vote
1answer
143 views

Approximate $n$ grade Bézier through cubic and/or quadratic Bézier curves

I'm trying to draw a $6$ grade (start point, $4$ control points, end point) Bézier curve but the API offers me only cubic and quadratic curves methods. Is there a way to split or approximate the $6$ ...
2
votes
2answers
1k views

Equation for control point distance for fixed-length cubic Bézier path (with specific constraints)

A particular Stack Overflow question asks how to construct a specific cubic Bézier path of constant length. I have experimentally determined the ideal distances of the control points from the nearest ...
3
votes
0answers
253 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
2
votes
1answer
609 views

Determining the result of Boolean shape operations on closed Bézier shapes

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 ...
-1
votes
1answer
444 views

Calculating points on the curve

I want to get the x and y coordinates of a curve..How can i do that... In the above image.Is it possible to calculate the intermediate points(one side) by knowing starting and ending point
1
vote
1answer
299 views

Frenet frame formula misunderstanding

The Frenet frame formula says that the first derivation of the equation $q(t)$ is my view: $$q'(t) = \verb|vec_view|$$ the cross product of derivation one and two $q' \times q''$ is ...
2
votes
4answers
1k views

Changing a bezier curve by dragging a point on the curve itself rather than a control point

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 ...
1
vote
1answer
120 views

Control points are off for “negative” vectors in a poly-line Bézier curve

I need to calculate the control points of a Bézier curve passing through N points where N > 2. I have been able to use the equations in this post to get close... but when "negative vectors" (the only ...
0
votes
1answer
670 views

Finding two Bézier control points given three points

My apologies if this is asked in the wrong spot, I believe that this problem has a fairly simple solution... but it is beyond me. Given three points (A,B,C) drawn at random, how do you figure out the ...
0
votes
1answer
55 views

How can I count waypoints between a curve?

I have curve that is drawn between point A and B. I want to divide this curve to 100 smaller waypoints. How can I determine what these 100 waypoints are as coordinates, when I only know points A and ...
1
vote
2answers
559 views

How can I learn de casteljau algorithm? (from calculus)

I'm an highschool graduate who is currently waiting for college. Meanwhile, I'm trying to do a little project by myself. (Computer stuff) And yesterday, I found that I needed to deal with something ...
3
votes
4answers
6k views

Casteljau's algorithm - practical example

I have a dataset with about 50 points (x,y) and I would like to draw a smooth curve that can pass as closer as possible on those points. I have heard about Casteljau's algorithm for splines but after ...
1
vote
3answers
1k views

Control Points of Bézier Curve?

Could someone give me a reason/proof why the control points do not lie on the Bézier Curve? Perhaps involving Bernstein Polynomials, if possible? Thanks!