Questions tagged [parametrization]

For questions on parametrization, the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

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1answer
963 views

Proving that second derivative is perpendicular to curve

How can I prove the following? $\gamma (t)$ is unit speed, $\dot \gamma(t) \not= 0 \Rightarrow \ddot \gamma(t)$ is perpendicular to $\gamma(t)$ I don't really see where a problem would arise when $\...
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1answer
217 views

Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$

I've got a problem calculating the volume of the top half of a pseudosphere. The pseudosphere is parametrised by $$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...
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1answer
118 views

Parametrization of tori in $\mathbb{S}^3$

A torus in $\mathbb{R}^3$ can be defined parametrically by \begin{aligned} x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\ y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\ z(\...
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2answers
277 views

Parametrization of $a^2+b^2+c^2=2d^2$

Is a complete parametrization of primitive solutions to the equation $$a^2+b^2+c^2=2d^2\qquad a,b,c,d \in \mathbb{Z}$$ known? A reference would be great. Solutions to $a^2+b^2=c^2$ give solutions ...
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1answer
728 views

Find the parametric form $S(u, v)$ where $a \le u \le b$ and $c \le v \le d$ for the triangle with vertices $(1, 1, 1), (4, 2, 1),$ and $(1, 2, 2)$.

Find the parametric form $S(u, v)$ where $a \le u \le b$ and $c \le v \le d$ for the triangle with vertices $(1, 1, 1), (4, 2, 1),$ and $(1, 2, 2)$. I am told that one parameterisation is $S(u, v) = (...
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0answers
777 views

Show tangent of evolute is normal to original curve

The evolute of a curve R(t) is the locus of the centers of curvature of the curve. Using the parametric formulas, show that the tangent to the evolute is normal to the original curve. I know that the ...
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2answers
103 views

Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedean spiral by $$x(n) = \sqrt{n}\cos(2\pi\...
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4answers
17k views

Parametrization for the ellipsoids

Can someone help to describe some possible parametrizations for the ellipsoid: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1?$$ I am thinking polar coordinates, but there may be the ...
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3answers
581 views

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
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4answers
91 views

Curve dense inside the unit circle

For $\alpha$ a real, irrational number, I have been to prove that any point $(x,y)$ such that $x^2 + y^2 \leq 2$ can be written as $$(x,y) = (\cos( u) + \cos( \alpha u), \ \sin( u) + \sin(\alpha u)) $$...
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2answers
469 views

Parameterising the intersection of a plane and paraboloid

Suppose we have the paraboloid $z=x^2+y^2$ and the plane $z=y$. Their intersection produces a curve $C$, and certain surfaces bounded by it, for example the disc $S$ which directly fills the area of $...
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2answers
249 views

Question on parametrization of the boundary of a rectangle in $\;\mathbb R^2\;$

I'm interested in constructing a unit normal vector on the boundary of a rectangle in $\;\mathbb R^2\;$ and so I found these steps: However I'm having a really hard time completing step 0! How can I ...
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2answers
190 views

Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
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1answer
3k views

Find the intersection of plane and sphere

If the equation of the sphere is $x^2+y^2+z^2=1$ and the plane is $x+y+z=1$, then how can the equation of a circle be determined from the equations of a sphere and a plane? and what is the parametric ...
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2answers
1k views

Calculating the surface of revolution of a cardioid.

I have the cardioid $r=1+cos(t)$ for $0\leq{t}\leq{2\pi}$ and I want to calculate the surface of revolution of said curve. How can I calculate it? The parematrization of the cardioid is: $$x(t)=(1+...
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0answers
90 views

Confusion with the Enneper-Weierstrass parametrization

The Enneper-Weierstrass parametrization is defined in terms of some complex integrals. Certainly if it happens that the function has some singularities, the integral may depend on the path. In ...
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0answers
32 views

Tori of revolution in $\mathbb{S}^3$

The motivation of this question is here, where I asked about some kind of general parametrization for tori in $\mathbb{S}^3$. Apparently not even the parametrization that I included in that question ...
2
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2answers
46 views

Finding limits of integration using spherical coordinates

I would like to integrate some function $f:\mathbb{R}^3\to\mathbb{R}$ over $C_1\cap C_2$ where $$C_1=\{(x,y,z)\in\mathbb{R}^3:x^2+4y^2+9z^2\leq1\}$$ $$C_2=\{(x,y,z)\in\mathbb{R}^3:x^2+4y^2+9z^2\leq 6z\...
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1answer
28 views

Proof of equivalence well defined function

This is the definition: Definition 1: A function $f:D\subseteq R\to R^n$ is said to be continuously differentiable of a $C^1$ function, if f is differentiable and the first derivative of f is ...
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1answer
4k views

Find a parameterization of the paraboloid

Find a parameterization of the paraboloid $900z = 25x^2 + 36y^2$. My Work $24x^2 + 36y^2 = 900z$ $\implies (5x)^2 + (6y)^2 = (30\sqrt{z})^2$ We can represent this equation using cylindrical ...
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1answer
120 views

Why, mathematically, does a closed curve parametrized by $\theta$ give the correct average of the distance between the center and perimeter?

The "average radius" is the average of the distance between the center and the perimeter of the closed shape. It "appears" correct that a curve parametrized by $\theta$ gives the correct average ...
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1answer
59 views

parametrical representations of polygons

Could you please explain, how one gets this Parametric representation of a solid trapezoid ? I mean the procedure and not the answer. I have some linear geometry (as polygons), and I need to represent ...
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1answer
42 views

Area cylinder limited by cone

I'm ask to find the surface area of the cylinder $x^2+y^2=2x$ limited by the cone $z=\sqrt{(x^2+y^2)}$ and the plane $z=0$ and . I know that the cilinder's center is at $(1,0)$, I understand how the ...
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3answers
51 views

Calculate the length of parameterized curve. (Lacking intuitive understanding of subject)

Problem Calculate the length of parameterized curve which is: $$ r(t)=(\frac{\sqrt{7}t^3}{3},2t^2)$$ in which $1 \le t \le 5$ Attempt to solve We can express our parameterized curve in vector ...
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1answer
217 views

Parametric curves with constant length differential

Given a 2D-curve arc $\mathscr{C}$, I would like to be able to easily compute a subset of $n$ points belonging to $\mathscr{C}$, so that the points are separated by equal-length curve arcs. For that ...
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1answer
219 views

Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
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1answer
1k views

Find a vector parametrization of the circle contained in the plane x=5 with radius 3 centered at the point (5,1,2)

I know the vector parametrization of the circle contained within the yz-plane centered at the origin is: $\vec r$( $\theta$ ) = < 0, 3 cos $\theta$ , 3 sin $\theta$ > What do I do next? If the ...