# Tagged Questions

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### Umbilic Points of an Ellipsoid

I have an ellipsoid given by $S = \{ (x,y,z): \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$, for some fixed $a,b,c \in \mathbb{R}^{+} \}$. I need to find the umbilic points of ...
All over the net it is stated that the parametrization of the lemniscate with Cartesian equation: $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$ is: $$\varphi: t \mapsto ... 1answer 54 views ### Two curvature formulas when equal arc-length all. So with a parametric curve \vec{r}=\langle x(t),y(t)\rangle, curvature is given by$$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$When we have constant arc-length, an alternate ... 3answers 84 views ### Parametrization of y^2 - x^2=1 I have found parametrizations for the level curve y^2-x^2=1, however, I have a question regarding one of them. From the Pythagorean trigonometric identity \cos^2 x + \sin^2 x =1 we obtain ... 1answer 54 views ### There exists a constant arc length parametrization I heard that for any curve in the plane that can be given parametrically by \vec{r}(t)=\langle x(t),y(t)\rangle for a\leq t\leq b that there exists a constant arc length parametrization, i.e. ... 2answers 86 views ### Extend a vector field of normal vectors beyond the surface I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in {R}^3 defined parametrically by \vec{r}(u,v) where ... 0answers 63 views ### Rheotomic surfaces parameterization? Are there parameterizations for rheotomic surfaces? Or, am I stuck with implicit formulas and marching cubes for plotting points? Are there special cases where the surfaces are parameterizable? Here ... 1answer 288 views ### Sphere parameterization with 6 patches I am looking for a parameterization of the sphere with 6 patches, like in http://www.image.ucar.edu/staff/rnair/research09.html and the inverse of this parameterization. As well, I would need a ... 1answer 41 views ### Existence of Lipschitz reparametrization Suppose we are given a continuous path,$$\gamma:[0,1]\rightarrow (X,d)\text{,}$$in a metric space (X,d). When we deal with differentiable enough paths in Riemann manifolds we can give a ... 1answer 106 views ### Tangent Vectors in a Surface As of recent, I've been studying Differential Geometry per the Dover Publication on the subject, and I've ran into a bit of an issue with tangent vectors to a parametric surface  \mathbf{x}(u^1,u^2) ... 1answer 75 views ### Is a parameterization defined to be surjective and/or injective? A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know ... 1answer 66 views ### 3D Surface parametrization basics I'm studying 3D rendering: I have a surface and the points on the surface are given by some function f such that p = f (u, v) Since I'm a newbie this is unclear to me: how can a function of u and v ... 1answer 119 views ### Parametric Equation of a Circle Using a Line Consider the unit circle$$ x^2+y^2=1. $$How can I parametrize it using the line y=m(x+1), where m is its slope? 1answer 142 views ### Parametrization of a solid Find a parametrization \sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3, with I a parallelepiped, of \lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace . 1answer 169 views ### variation problem of constrained area and minimized distance$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$The integral is a time-like curve between x_1 and x_2 and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ... 3answers 231 views ### Curvature of the image of a curve projected onto a surface (Adding a bounty since I need more details than I have so far) Given a point$$ s_{0}=S(u_{0},v_{0}) \;\;\;\; (S:\mathbb{R}^{2}\to\mathbb{R}^{3}) $$and a point$$ c_{0}=C(t_{0}) \;\;\;\; ...
The n-dimensional cone, with vertex at the origin, central angle, $\alpha$ and central axis in the direction of the unit vector $\xi$ is defined to be all those points, $x\in {R^n}$ whose dot product ...