The strong convex hull property says that for a spline of degree , the convex hull of at most control points of a spline contains the point of a spline at parameter . Intersection methods can use this property to rule out certain interections. Union of a convex hulls of consecutive points completely encompass the … More sandwiching a spline
In the post on tessellation step length where we discussed ways to get a bound for step length, the trickiness of computing maxima forced us to take a different approach to getting a bound. We had to look at integrals as alternatives. In this post we take a closer look at computing points where maxima … More tightly bounding a bspline
A common step in graphics processing and in geometry processing is to is to discretize(tessellate) curves into smaller segments that closely approximate the curve. Let be a curve defined on . In this post we will compute a value for global step size such that the chord from to is not farther from than for … More Tessellation step length
Computing arc length, and arc length parametrization using a queer looking power series … More arc length
Last year I worked on a interesting problem which asks about the possibility of partitioning a convex set in a plane in to internally disjoint convex pieces of equal area and perimeter. The problem has since been proved when is a prime power, by Aronov and Hubard and Roman Karasev for all dimensions. Here in … More Waist on a torus
The binomial like coefficients that we saw in earlier posts: are called –binomial coefficients. In keeping with the existing convention we will use as the indeterminate instead of and write for It was established in an earlier post that: Using this identity we may reduce the -binomial coefficient to: We also made use of the … More q-binomials
This post culminates our efforts in solving the problem: If are positive integers with and satisfying the following is divisible by for all positive integers , then is a power of . First recap a few facts from this post: We began by showing that: is divisible by for all and therefore by the lcm, … More ponder problem now on last legs
In our build up to a proof of this problem., we established that the product is a multiple of each of the numbers for . But we wished to show the stronger result that is divisible by . One way to proceed to estimate how the lcm, differs from the product and see if also … More another nugget concerning cyclotomic polynomials
In this post we record a useful result concerning cyclotomic polynomials. This will help us prove what is claimed in this post. Theorem: For distinct positive integers the GCD of the values of cyclotomic polynomials and , is only if for some integer . On the other hand, if for some , . Proof: () … More a result concerning cyclotomic polynomials
It is well-known that if natural numbers and satisfy , for some , then is a multiple of for all ,. April 2005 ponder asks whether the converse is true, that is if for integers , the number is a multiple of for all , then a power of . While trying to prove this … More an old ponder problem
Bedtime mathematics 