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.

27 questions
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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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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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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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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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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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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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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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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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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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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 ...
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 ...