How to get a tangent vector at bezier point

#1

Hi All!
I´m writing a bezier interpolation method on c++.
The Problem is :
I can get a point position for the parameter (t), but i dont know how i can get a tangent vector for this point.
Any ideas?

#2

Just differentiate the parametric interpolation function with respect to t?

#3

Can u get me a formula?

for example I have a 5 arguments

POINT start;
POINT end;
VECTOR tangentStart;
VECTOR tangentEnd;

double t;

Now I´ve a some method to get a outPoint:

POINT bezierInterpolate(POINT &start, 
  					  POINT &end, 
  					  VECTOR &tangentStart, 
  					  VECTOR &tangentEnd,
  					  double t)
  {
  	// bla bla ... 
  	return pointPosition;
  }

I would change this code to, for example, this

struct parameterPoint{
  	  POINT position;
  	  VECTOR tangent;
  }
  
  parameterPoint bezierInterpolate(POINT &start, 
  					 POINT &end, 
  					 VECTOR &tangentStart, 
  					 VECTOR &tangentEnd,
  					 double t)
  {
  	// bla bla ...
  VECTOR tangent  =   WHAT I NEED TO WRITE HERE????????? 
  	return parameterPoint;
  }
#4

Well here is a nice discussion of cubic Beziers

The coefficients a, b, and c are derived from the four points ( which you can get from your endpoints and tangents). Then just differentiate the parametric polynomial to get
p’(t) = 3at*t + 2bt + c

#5

So, depending on how many tangent computations you need for a given Bezier curve, it may be useful to compute a hodograph. The idea of a hodograph is that you create a curve which when evaluated at the same t value as the original gives you the tangent.

Alternatively, there are some good speedups and simplifications that are possible if you’re computing at the endpoints of a curve. If you’re not computing tangents often, it may be simple to split the curve at your t value, and then compute the tangent at the end of that new curve.

The simplest method is just to compute the point position at t, and then at t + epsilon (where epsilon is small), and get a vector that way. Not particularily fast or accurate, but easy, and might give you what you need.

Details and formulas available here:
http://www.tsplines.com/resources/class_notes/Bezier_curves.pdf

Let me know if you need some pointers… I’ve implemented hodographs and the tangents at endpoints, I bet I can dig the code up somewhere.

#6

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