I am trying to find the intersection point of a curve and a 3D surface with no luck. The surface is in the shape of a cone, and the curve is hyperbolic, as are shown in the figure. CONE AND THE CURVE
This simulates a ray hits a certain surface. I tried to use bisection method, but it doesn't seem to work. then I tried newton's algorithm, but the results are still not good.
Is there any other good algorithms out there which are suitable for solving this kind of problem?
You are searching for a curve-surface intersection algorithm. Note that both curves and surfaces can be represented in either implicit form or in parametric form. Surface in implicit form is defined by equation F(x, y, z) = 0, which is a quadratic polynomial of x, y, z in case of conic surface. Surface in parametric form is defined by point-valued function S(u, v) of its parameters (e.g. you can use distance along cone axis and polar angle as parameters of conical surface). Curve is usually described only in parametric form, as a function C(t) with parameter t, which could be quadratic for a hyperbolic curve.
The simplest cases of all is to treat your problem as an intersection of parametric curve against implicit surface. In this case you can write down a single equation q(t) = F(C(t)) = 0 with single variable t. Of course, Newton's iteration is not guaranteed to find all solutions in general case, bisection can only surely find one solution if you find two points with different sign of q(t).
In your case q(t) is a quartic polynomial (after putting quadratic curve parametrization into quadratic surface equation). It can be theoretically solved with Ferrari's analytic formula, but I strongly advise against it, because it is quite unstable numerically. You can apply any popular polynomial solver here, like Jenkins-Traub algorithm or eigenvalues algorithm for companion matrix (also see this question). You can also use methods of interval mathematics: for example, you can recursively subdivide the domain interval of parameter t into smaller pieces, while pruning all the pieces that surely do not contain zeros (interval arithmetic would help you to detect such pieces).
Now we can move on to the case when both the curve and the surface are represented parametrically. I do not know any solutions that could benefit from the fact the your surface is conical and your curve is hyperbolic, so you have to apply the general curve-surface intersection algorithm. Alternatively, you can fit an implicitly-defined cone into your parametric surface, then use the solution above for quartic polynomial roots.
A lot of reliable general intersection algorithms are based on the subdivision method (which is actually interval mathematics again). The general idea is to continiously divide the curve and the surface into smaller and smaller pieces. The pairs of pieces which surely do not intersect are dropped as soon as possible. At the end you'll have a set of small piece pairs, tightly bounding your intersection points. Yoy might want to run Newton's iteration from them in order to make intersection points precise.
Here is the outline of a sample algorithm:
For each pair of curve piece and surface piece, you have to check whether they can potentially intersect, which can be easily done by checking their axis-aligned bounding boxes. Also, you can represent your curves and surfaces as NURBS, in which case you can use convex hulls as tighter bounding volumes.
Generally, there are tons of variations and improvements of this algorithms. I advise the following literature for deeper knowledge:
If you are seeking for a simple and working solution, and you are sure that hyperbolas and cones are the only things you have to worry about, then you'd better use implicit definition of cone and solve quartic equation with some standard numerical algorithm from a good library available to you.
