1
vote
1answer
107 views

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 ...
3
votes
2answers
131 views

Parametrization of the lemniscate

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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
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. ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
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) ...
1
vote
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 ...
0
votes
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 ...
2
votes
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?
1
vote
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 $.
0
votes
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 ...
6
votes
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}) \;\;\;\; ...
2
votes
1answer
183 views

Parametric Equations for a Hypercone

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 ...