## Shortest path in the plane under derivative constraint November 27    1

A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0.This is of course very close to the classic introductory p

## Infimum length of curves November 27    1

Let the unit disc $\{(x,y): r^2=x^2+y^2<1\}\subset\mathbb R^2$ be equipped with the Riemannian metric $dx^2 +dy^2\over 1-(x^2+y^2)$. Why does it follow that the shortest/infimum length of curves are diameters? I remember doing an optimization course

## Shortest path on hyperboloid November 27    4

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the shortest path between two points on the surface? An

I have some problem in the calculation of the derivative $\frac{dy}{dx}$ of the following parametric functions: $$x(\theta)=a\cos(\theta)\left[ 1-\frac{1}{2}\cos(\theta)^2\right],y(\theta)=\frac{1}{2}a\sin(\theta)\cos(\theta)^2$$ I tried to obtain $\ ## Calculating the length of a curve for$ y=\ln (1-x^2) $November 27 1 In calculating the length of$ y=\ln (1-x^2) $on the interval$ [0, {3\over4}]$, I found that for whatever reason, I end up with invalid domains in the indefinite integral using the given l ... ## When trying to calculate arc length, what is the easiest way to approach the$(dy/dx)^2$portion November 27 1 When trying to calculate arc length, what is the easiest way to approach the$(dy/dx)^2$portion?If I have: $$x = \frac{1}{3}\sqrt{y}(y-3),\qquad 1\leq y\leq 9;$$I take the derivative of the function and get$\frac{1}{2}y^{1/2} - \frac{1}{2}y^{-1/2}$## Finding a polynomial$F$such that$F(P,Q) = 0$, where$P$and$Q$are polynomials. November 27 1 I am a bit struggling with this at the moment :Let$K$be a field and let$P, Q \in K[X]$.Is there always a (minimal?) polynomial$F \in K[Y,Z]$such that$F(P,Q) = 0$?And if/when the answer is positive, how to find such a polynomial?For example, wit ## conjecture regarding the height of polynomial's square-free part November 27 1 About some time I am struggling with the following interesting problem:There is a well-known theorem of Mignotte which says that for a polynomial$f\in\mathbb{Z}[x]$of degree$n$and height (coefficient size)$2^\tau$, the height of its divisors is ## Find the square root November 27 2 My question is:Find the square root -$(x-1) (x^3 + 4) + (\frac{x}{2} + \frac{2}{x})^2$The above is a polynomial.I would like to how to find its square root.Not for all such expressions there exists a "root", but if it does, you can try this way: ## What$t$coefficient should I choose for a Bezier curve November 27 1 For a cubic Bezier curve, I have this formula: $$\mathrm{B}(t)=\mathrm{P}_0(1-t)^3+3\mathrm{P}_1t(1-t)^2+3\mathrm{P}_2t^2(1-t)+\mathrm{P}_3t^3,\ t\in[0,1]$$Now about$t$I only know that is should be between$0$and$1$, but which value should I choo ## Showing that Bezier curve length is less than its control polygon November 27 1 This is a homework and pardon me for the huge gap of my Mathematics knowledge. After thinking and referencing for a few days I came up with something like following, appreciate help to comme ... ## Rotate a line segment with an angular constraint to fit exactly between two rays in 3d space November 27 Hi my Mathematics are somewhat rusty and I am trying to solve a problem where I take a 3d line segment described by the vector r*(cos[t], h, sin[t]) where t is unknown and describes a constraint that will rotate its end points to exactly fit in-betwe ## segment intersecting a tetrahedron November 27 1 I am trying to write C++ code to find the intersection points of a segment intersecting a tetrahedron. I reduced the problem like this:For each face of the tetrahedron (a triangle), find the intersection point of the line segment. Then, I have three ## intersection of an ellipsoid and cylindrical plane. November 27 1 I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an cylinder is x^2+y^2=R^2, z=height and equation of an el You Might Also Like • Help me please with this question.Let$f\in C^{\infty }$function defined as$\forall x, f(x)=f(x+2\pi )$. L ... • how to solve$z^2 +3|z| = 0 , z$complex ? treating the complex number as$a+bi $or anything similar didnt ... • A linear transformation, $$T: \Bbb{R}^m \rightarrow \Bbb{R}^n$$ is a function that has the following propert ... • Let$k$be an algebraically closed field. Let$I$be the map that takes algebraic sets in$k^n$to the ideal ... • I know a proof of the following theorem using determinants. For some reason, I'd like to know a proof withou ... • Question: Prove that the diameter$\mathcal p(S)$of a simplex$\mathcal S$equals the greatest Eucledian di ... • Let$A$be a skew-symmetric$n\times n$-matrix over the real numbers. Show that$\det A$is nonnegative. I'm ... • How to find the divergence and the curl of the given vectors?a.$( \vec{u} \cdot \vec{r}) \vec{v}$b.$( \vec ...
• let $(X,d)$ and $(Y,\rho)$ be complete metric spaces and let $f\space:\space X\rightarrow Y$ be a surjective ...
• Reading on control theory and the Laplace transform of the unit step function, I came upon the following in ...