Hermite/quadratic bezier curve vs (sin and cos)

[arpace]

So I have a Hermite curve with the control points:

//point# (x,y):

p0 (1,0)
p1 (1,(Math.sqrt(2)-1))
p2 (Math.sqrt(0.5), Math.sqrt(0.5))

I also have the algorithm to find the points on the locus between 0 and 45 degrees:

//L0 is the point on the line between p0 and p1
//L1 is the point on the line between p1 and p2
//Loc is the point on the locus
//...
//keep in mind that although this can be extended with if statements to check for
//any degree with a little modification
//right now, the rule is 0 <= degree <= 45

L0 = p0 + (p1 - p0)*degree/45;
L1 = p1 + (p2 - p1)*degree/45;
Loc = L0 + (L1-L0)*degree/45;

so Shouldn't this, as it is essentially a quadratic Bezier (Hermite) curve representing degrees 0 through 45 of a unit circle , not equal the results of sind(degree) as long as:
0<= degree <= 45 ?


when I use for the degree 22.5, on my windows calculator
sind(22.5) = 0.3826834323650897717284599840304;

...but mine (again using the window calc) at 22.5 degrees is
Loc.y = 0.38388347648318440550105545263106

could this be due to floating point error? Is there something obvious I am missing?

here is an image I made to help with conceptualization
[PLAIN]http://sphotos.ak.fbcdn.net/hphotos-ak-snc4/hs1384.snc4/163638_10100121877154282_28122817_59279302_5323393_n.jpg

 

[arpace]

seriously? no one?

 

[AlephZero]

A Bezier curve can not be exactly equal to a circle.

Repeat the subdivision process again, and you will see that you a mid-point which is not a unit distance from the origin.

However your splines are a good approximations to a circle, and are often used to draw approximate circles in computer graphics.

FWIW it is possible to represent a circular arc up to a semicircle (or in fact any conic section) exactly using rational functions as splines. Look up the theory of NURBS (non-uniform rational B-splines) if you are interested.

 

[timthereaper]

Actually, a quadratic rational Bezier curve can exactly reproduce a circular arc and a degree 5 rational Bezier curve can represent an entire circle. There is a formula to figure out the weights for each control point if you're still interested. A NURBS can reproduce any conic section, which makes them so versatile. However, a NURBS is basically a collection of Bezier curves joined together so that they have a specified continuity. If you're interested, there is a whole bunch of information about computer-aided geometric design at http://en.wikiversity.org/wiki/CAGD

 

[CellCoree]

I would like to first clarify that both the Hermite/Quadratic Bezier curve and the sine/cosine functions have their own unique properties and applications. While they may both represent curves or points on a locus, they are not necessarily equivalent.

With that being said, let's address the specific question at hand. The algorithm provided for finding points on the Hermite curve between 0 and 45 degrees seems to be correct. However, it is important to note that this algorithm is only valid for the specific control points given. If we were to change the control points, the resulting curve may not match the results of the sine/cosine functions.

Furthermore, there may be slight discrepancies between the results of the algorithm and the results of the sine/cosine functions due to floating point error. This is a common issue in computer calculations and can be minimized by using higher precision data types or by implementing error correction methods.

In conclusion, while the Hermite/Quadratic Bezier curve and the sine/cosine functions may have some similarities in representing curves or points on a locus, they are not interchangeable and their results may not always match due to differences in their underlying algorithms and properties. It is important to carefully consider the specific application and intended use before choosing one over the other.