User maritza sirvent - MathOverflow

Answer by Maritza Sirvent for Interesting Calculus Questions/Exercises

2011-08-08T16:41:21Z

For a project in calculus, I like the problem of the brachistochrone, the students need to investigate about the curve that minimizes the time that takes a bead rolling down from a point to another one not right below the first one. They also are asked to do a prototype of the curve. A calculus book that I like, and use in my courses is the one by George F. Simmons (Calculus with Analytic Geometry)

Answer by Maritza Sirvent for Finding two bezier control points given three points

2011-08-08T01:08:54Z

I am not sure I understand your question, but if you want the set of points that will describe the quadratic Bezier curve defined by the three points A, B and C, try the DeCasteljau algorithm. That is, you go halfway the points A and B, then halfway the points B and C, then join these two new points, then go halfway these two new points, and the point you obtain lies in the quadratic Bezier curve .. and you do this for every value on the interval [0,1], to construct the whole quadratic curve.

Once differentiable, piecewise degree three polynomials on triangulated planar domains

2011-08-05T16:21:22Z

Here is an easily described, but very difficult, problem that I (and a number of other people) really would like to see solved during our life times. The basic problem is to compute the dimension of a certain vector space. Suppose we are given a triangulation (a tessellation by triangles where any two triangles share at most a common edge or a common vertex) of a polygonal domain in the plane. Let $S$ be the space of once differentiable functions on that triangulation that on each triangle can be represented as a bivariate polynomial of degree 3. $S$ is clearly a vector space. It is known that the dimension of that space is greater than or equal to $3V_B + 2V_I + 1 + \sigma$ where $V_B$ is the number of boundary vertices, $V_I$ is the number of interior vertices, and $\sigma$ is the number of singular vertices of the underlying triangulation. A singular vertex is an interior vertex that has exactly four edges attached where those edges form two parallel pairs. (In other words, a singular vertex is the intersection of the diagonals of a convex quadrilateral.) It is known that generically the dimension of $S$ equals the given expression, and there is no case known where the dimension is larger than that expression. Many people conjecture that the lower bound equals the dimension for all triangulations. Prove, or disprove, that conjecture.

Some background: triangulations are the natural generalization of a partition of an interval to two variables, and $S$ is a spline space with potential for a wide range of practical problems, such as data fitting or solving partial differential equations. The problem has been known among approximation theorists since the early 1970s, and despite efforts by a number of people the problem is still unsolved. For more information on spaces like $S$ see the recent book by Lai and Schumaker [Spline Functions on Triangulations, Cambridge University Press] in particular section 9.9. The basic issue with the kind of spline space considered here is that the dimension depends not just on the topology of the triangulation, but also on its geometry. An arbitrarily small change of the location of the vertices can change the dimension. If the polynomial degree is four (instead of three) it is known that the dimension can change only when a vertex switches between being singular and non-singular. On the other hand, if the polynomial degree is two (instead of three) many configurations other than singular vertices are known where the dimension changes with the geometry. So the polynomial degree three straddles the boundary, and we would like to know to which camp it belongs.

Comment by Maritza Sirvent

2011-08-06T04:15:02Z

Noam, You are right, it should read greater than or equal to ..

Comment by Maritza Sirvent

2011-08-05T21:42:17Z

Dear Ryan: Thank you for editing the title .. I will edit the way I wrote this open question so it fits better into the spirit of this platform ..