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3

It's not hard to determine parametric equations for the surface. We have $$x(t) = t, \qquad y(t) = t^2, \qquad z(t) = t^3,$$ so $$x'(t) = 1, \qquad y'(t) = 2t, \qquad z'(t) = 3t^2;$$ therefore, the tangent line to a point $(t, t^2, t^3)$ is $$(t, t^2, t^3) + s(1, 2t, 3t^2),$$ and consequently the surface $S$ consisting of all such lines is parametrized by ...


2

Let $0 < r < R$ be real numbers. On the circular torus parametrzed by $$ X(u, v) = \bigl((R + r\cos u)\sin v, (R + r\cos u)\cos v, r\sin u\bigr), $$ the coordinate vector fields and their cross product are \begin{align*} X_{u} &= (-r\sin u \sin v, -r\sin u \cos v, r\cos u) \\ &= r(-\sin u \sin v, -\sin u \cos v, \cos u), \\ X_{v} &= ...


2

I know two approaches : Bilinear filtering Minimal surface The first is very simple but not that nice looking. The second is much more beautiful but not hat easy to implement. The picture looks like to be about a minimal surface.


1

A point on the surface is any such point $(x,y,z)$ for which $2(x-1)^2 + (y+2)^2 + z^2 = 2$, as that is the definition of the surface. To find a single point on the surface, plug in a couple of values into your equation and see what you get. For example, $(1,1,1)$ is not on the surface because $2(1-1)^2 + (1+2)^2+1^2=10\neq 2$ OK, let's see, maybe a ...


1

(Since you didn't specify, I'm assuming that the distances $r_1, ...$ are distances on the surface, not in $\mathbb{R}^3$. Sorry if this turns out to be useless!) Draw the lines between the centers of the circles. This divides the sphere into four spherical triangles -- making it a "spherical tetrahedron", if you will. We know the side lengths of the ...


1

The term "Coons patch" is used to refer to several different surface types in CAGD, unfortunately. It sounds like you're talking about a standard tensor-product bicubic patch, which can be written in Hermite form: $$ \mathbf{S}(u,v) = [1 \; u \; u^2 \; u^3] \cdot \mathbf{M} \cdot [1 \; v \; v^2 \; v^3]^T $$ where $\mathbf{M}$ is a $4 \times 4$ matrix ...


1

In the third equality of the Gaussian curvature calculation, you appear to have $\det(\frac{1}{r}A) = \frac{1}{r} \det(A)$. But $A$ is $2 \times 2$, and $\det(cA) = c^{n} \det(A)$ if $A$ is $n \times n$. :) Separately, $df_{f(p)}^{-1} = \frac{1}{r} A^{-1}$ (rather than $\frac{1}{r}A$). This doesn't affect the outcome of the Gaussian curvature computation, ...


1

Here's an approach: for problem 2, pick specific values for $\mathbf n$ and $b$, like $$ \mathbf n = [0,0,1]^t \\ b = 2 $$ and see what the result looks like. Then see whether you can generalize. The same approach should work for all the others.


1

Beltrametti, Carletti, Gallarati, Bragadin Lectures on Curves, Surfaces and Projective Varieties: A Classical View of Algebraic Geometry p 227-8: Let $C$ be an irreducible non-planar cubic in ${\Bbb P}^3$ [..]for each point of ${\Bbb P}^3$ there passes one and only one chord (or a tangent) of $C$ , and write the equation for the ruled surface of the ...


1

If the relative volume of two similar objects is 20,000 , then the relative "lengths" of the two objects is the cube-root of 20,000 (because volume has three dimensions, so you have a product of three lengths). The relative surface area is then the square of that result, since area is a product of two lengths. Since square meters and square centimeters are ...


1

There are three ways I like to think about surfaces: 1) as graphs of functions from $\mathbb{R}^2$ to $\mathbb{R}$, 2) as level sets of functions $\mathbb{R}^3 \rightarrow \mathbb{R}$, and 3) as images of maps $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Surfaces of the third type are called parametric. Here are examples of each: 1) The graph of $z = x^2 + ...


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$\def\RR{\mathbb{R}}$I had trouble extracting this from Struik, probably because I don't know enough classical facts about envelopes of families of planes, so I'll write up my solution. I'll say more about how Struik confuses me below. (On rereading your question, I was confused at an earlier point than you: I didn't get why everywhere flat implied $X$ ...


1

This doesn't exactly answer the question, but with $k$, $m$, and $n$ positive integers, the parametric equations \begin{alignat*}{3} x(s, t) &= a\cos(mt) \cos^{k}(ns) &&\cos(t) &&\cos(s), \\ y(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(t) &&\cos(s), \\ z(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(s) && ...


1

It is perfectly fine. % $ \text{area occupied of rectangle }R_2 \text{by rectangle } R_1 = \frac{\text{area of} R_1}{\text{area of} R_2} \times 100$


1

$\def\RR{\mathbb{R}}$I finally managed to do the routine computation Ted Shifrin describes, and I'm writing up the details for the record. It turns out that the Gauss map is a bit of a red herring. Rather, let $X$ and $Y$ be two surfaces in $\RR^3$ and let $f: \RR \to X$ and $g: \RR \to Y$ be two curves which have the following interesting property: For ...



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