

Bezier curve and its parametric equations have been proved as an effective and efficient math tools to describe and simulate any piece of curved segment and used for computerization as a modern design and animation tool. However, the intersection points problem is also the development bottleneck of the contemporary 2D/3D CAD/CAM tools as there were no effective solution to solve them.
My solutions are not only solved the Quadratic and Cubic Bezier curves' X points problems to reach the contemporary requirement, but I have also pushed its ability into the next degree, QuarticBez, as well. One degree higher means one more level of continuity and a higher level of controllability and the better simulation ability to much more complicated curve form. Also I believed that CubicBez and QuarticBez will be used to cover most of our current and future of design and simulation requirement.
Also, the best things of my solution are, it is based on geometry theory and it is real time operable in any modern PC.
The solutions that I have done:
 I have solved all the intersection points in between two degree 2, degree 3 and degree 4 of Bezier curves (or socalled Bernstein curves); and more …. Please find them in the pages of “X of Curves”.
 I have solved all of the SelfX points of degree 3 and degree 4 of Bezier curves.
 I have proved the Identity
problems of QuadraticBez and CubicBez in math, which are about the questions of if any of two Quadratic Bezier curves belong to one parabola or not? and if any of two Cubic Bezier Curves belong to a single degree 3 curve or not? Now, I am still working
on how to identify any of two different QuarticBez segments are picked from the same piece of degree 4 curve or not?
The features of my solutions:
 My solutions are all general and common to solve the above problems, they are not limited for special cases only.
 My solutions are all realtime operable in any modern PC; which means that they are fast and should be the fastest in the whole world now.
 The kernel ideas of my solution are all based on algebra and geometry theories. Though I do need two approximation algorithms to run my solutions, but the first one is the PolySolver that help me to solve the polynomials in my equations and the other is developed by myself to be used to compensate the precision loss for computational roundingerrors during the general calculation and the prosess of polynomial roots finding.
 The calculated results of X point from my solution are very precise, and the precision level can be controlled by setting the threshold value in program. The current threshold for the X (intersection) points in my demo program is set at 1.0e9.
 My solution is not just fast and precise only, it is pretty stable too. There is no unfound X point.
Note:


Key advantages of my solution: Accuracy, Precision & RealTime Speed
People asked me about what is the difference of my solution from the others recently? How do I speedup my algorithm for realtime operation? In return, I would like to ask one question too. How do we solve the intersection problem in between a circle and a parametriccurve? I believe many people would say it is pretty simple ... Why? What is the difference of this problem compared with the problem of two curves that intersect with each other?
The difference is, in the former problem we have to deal with one clock only as the circle is in a constant world without time changing, but in the latter one we have to deal with two clocks that tick asynchronously most of the time. So if I just turn off one clock in the two curves then the complicate problem would suddenly changed itself into a simple one, right? I guess this is also the secret reason why, I am the only person in the whole world so far that can solve the selfintersection problem of cubic and quartic Bezier curves in realtime as well.
Why the “INTERSECTIONs” are so important with curves?
It is simply because without knowing how to solve them then we can not calculate the union or intersection of any two 3D curved objects. Also without these calculated results, we are not able to turn an 3D curved design into the machine tool paths then we have no way to produce the object with our modern tooling machines.
Bezier curve equations are actually a group of descriptions of nature curve segments in different degree; and curves are actually the moving behavior of a combination of vectors of force. Also Bezier equation is super fit into our modern computer calculation and drawing technologies for curved object design or curved model building. All these reasons make it an unreplaceable tool. However just because we were stuck on the problems of intersections in between curves and within themselves so its truly usage is very limited so far.
In
a summary, it means that though we knew this tool is fantastic to
describe and operate CURVES; but if we do not know how to conquer the
cross points problem of any two parametric equations then we would have
big trouble to control and predict their mutual interactions.
Your solution seems 2D only, how does it related with the 3D problems?
It is true, I have presented all of my data examples in 2D here, but it is also a 3D solution without question. Because of that any 3D Bezier curve is easily projected into a plane to get its 2D curve part, and combine two plane solutions like XY and YZ back to 3D is just a piece of cake. So turning the 2D answers into 3D combination is a simple process just like solving a problem of two associated equations in one by one order.
Actually
the PROJECTION way is a kind of professional talking, if you want to
turn a 3D Bezier into 2D one, just drop the z part of coordinate from
those control points then you get its 2D curve part on XY plane from
the original 3D one; simple, right?
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