# 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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### Parameterization of Curves and Surfaces

I would like to know if the following settings are correct: a) Parameterization of Curve: Given the curve $C = \{(x, y, z) \in \mathbb{R}^3 \,|\, x^2 + y^2 = e^3\}$ I want to find a parameterization ...
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### Parametrising a piecewise circular curve in 3d

The curve in question is a closed curve $C: = \overline{abcda}$ made of three circular arcs $C_1=ab, C_2 =bc$, $C_3 = cd$ and a straight line segment $C_4 = da$. The arc $C_1$ lies on a plane $P_1$ ...
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### Explanation for arc length in parametrized curve

let $\Delta s_i$ be a piece of arc length hence: $$\Delta s_i = \int_{i-1}^i \sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}dt$$ Why is that the length of $\Delta s_i$ in 2d I know that the as $\Delta x$ -> ...
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### Preimage of a point in differential geometry of surfaces.

I'm studying how to work with surfaces in differential geometry. The definition of a regular surface is the following one: A subset $S\subset\mathbb{R}^3$ is a regular surface if, for each $p\in S$, ...
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### Finding x-axis-intercept of a parametric equation and slope

let $x = t^2$ and $y = t^3-3t$ now the equation for the slope is: $$\frac{dy}{dx} = \frac{3t^2-3}{2t} = 0$$ now at the point (0,0) is the first intercept with the x-axis with slope $\infty$ what is ...
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### Parametric solution of quartic diophantine equation in three variables

How can I handle the quartic diophantine equation in three variables $x$, $y$ and $z$ $$x^4-x^2=y^2-z^2$$ in general, i.e, does exists a (three-variable) parametric solution? What I've tried is ...
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### Second derivate of parametric equations (Intuitive) [closed]

let: $$x=f(t)$$ $$y=g(t)$$ hence: $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$ How can I derive the formula for the second derivative of a parametric equation?
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### The curve $y^2=x^3+Ax+B$ can be parameterized if and only if $x^3+Ax+B$ has a repeated root.

I'm working through Shafarevich's Basic Algebraic Geometry, and one of the problems asks the reader to prove the problem in the title. I found the "if" direction fairly straightforward, but ...
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### Parameterizing the equation of a line in 2-D

I'm attempting to read Emil Artin's little book on the Gamma function with a borderline adequate background at best. In order to make it past the first page I need to understand what a convex function ...
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### Finding every solution of $a^2+b^2+c^2=3$ in $\mathbb{Q}(i)$

Specifically, $a,b,c\in\mathbb{Q}(i)$ are complex numbers with rational parts whose squares sum to $a^2+b^2+c^2=3$. There's an answer to this question over $\mathbb{Q}$ already here but I couldn't ...
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### Why are the singular values equal to the first partial derivatives.

I am studying computer science so please go easy on me. I am also too bad at math to extract the mathematical essence that is needed to answer this question so I'm just gonna explain the whole setup. ...
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### Accounting for "Glued" Edges in Finding Boundary Curves of Parametrized Surfaces

Let $T: [0,2] \times [0,2\pi] \to \mathbb{R}^3$ be defined by $T(r,\theta) = (\cos\theta, \sin\theta, r)$, which parametrizes a cylinder with radius $1$ and height $2$. To find the boundary curve of ...
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### equation of circle and ellipse to parametrize an arc

an equation of an arc is defined by the equation of a circle: $$(x-a)^2+(y-b)^2=R^2$$ so $$x=x(y)=a\pm \sqrt{R^2-(y-b)^2}$$ I want to be safe from using the $\pm$ solutions, so I thought if we can ...
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### Dodgy limit swap

Suppose one has $$F(b)= \int_{0}^{\infty} f(x, b)e^{g(x, b)}dx.$$ Next, suppose that $F(0) = \infty$ and $g(x, 0) = 1.$ Further suppose that $\lim_{b \rightarrow 0}bF(b)$ converges. Finally, let  ...
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### Why did the author used the intersection of the line $y = tx$ with the curve to find the parametrisation?

I want to parametrize the curve $y^2=x^2+x^3.$ When I was reading the textbook by Theodore Shifrin "DIFFERENTIAL GEOMETRY:A First Course in Curves and Surfaces". I found the parametrisation ...
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### Family of curves sharing the same length

Do you have an example of a family of curves $C$ that share the same length $L$? By family, I mean a set of curves that can be expressed in a generic form - using one or multiple parameters. Put ...