# Tagged Questions

For questions about parametric equations, their application, equivalence to other equation types and definition.

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### how to logically modify a variable so that it accurately fits a given curve

I have 2 sets of experimental data. Each set has 2 variables (A,B) and response data (C). A1 100 100 100 100 100 100 100 B1 11.3 10.1 8.9 8.1 7.7 6.5 5.3 A1/B1 8.8 9.9 11.2 12.3 13.0 ...
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### Complex integral: how to parameterise a circle?

Evaluate $$\int_\gamma \bar{z}^2dz,$$ where $\gamma$ is the circle with centre $1$ and radius $1$ traced anticlockwise. One parameterises the circle $\gamma$ as $z=1+e^{it}$ for $t\in[0,2\pi]$ and ...
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### Find a parametrisation of the arc of the circle with radius $r$ centred at $z_0$ between $\phi$ and $\theta$.

where $-\pi \leq \theta < \phi \leq \pi$. Im not sure how to start with this question.
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### Finding the Euler parametrization of a curve

I have the following question as a homework problem for my differential geometry class: find the curvature and the explicit Euler parametrization of the ellipse $\gamma(t) = (a \cos t, b \sin t)$ ...
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### Parametrizaction of a Hyperboloid

I do not understand why when you revolve a hyperbola around a circle the respective parameters (cosh (v) and cos (u)) are multiplied by each other to get the parametric form of the hyperboloid. I ...
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### If $a^2 + b^2 = 1$, show there is $t$ such that $a = \frac{1 - t^2}{1 + t^2}$ and $b = \frac{2t}{1 + t^2}$

My question is how we can prove the following: If $a^2+b^2=1$, then there is $t$ such that $$a=\frac{1-t^2}{1+t^2} \quad \text{and} \quad b=\frac{2t}{1+t^2}$$
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### Calculus problem of finding the equation of a line.

Find the equation of a line that passes through the origin, with positive slope, and its tangent to the parabola given by :$y = x^2 - 2x + 2$ My approach to this problem was to differentiate the ...
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I am trying to create a parametrized curve. Basically I want a monotone curve through $(0, 0)$ and $(u, 1)$ with a plateau at $(ru, s)$ with $u\gt 0$ and $r,s\epsilon[0; 1]$, so my constraints are $... 1answer 34 views ### How to represent 2D lines (i.e. on$x$and$y-axis$) on a 3D graph using either Cartesian ($z=f(x,y)$) or Parametric$(x,y,z)=f(u,v)$How to represent 2D lines (i.e. on$x$and$y-axis$) on a 3D graph using either Cartesian ($z=f(x,y)$) or Parametric$(x,y,z)=f(u,v)$Hi, I've been working on a Simplex problem and would like to ... 3answers 36 views ### Find the derivative$dy/dx$from the parametric equations for$x$and$y$Let \begin{cases} y=2t^2-t+1 \\ x=\sin(t) \end{cases} find$\frac{dy}{dx}$Is this all that I need to do? $$\frac{4t-1}{\cos(t)}$$ 2answers 39 views ### Parametric Equations. Find$\frac{dy}{dx}$in terms of$x$Find$\frac{dy}{dx}$in terms of$x$if the parametric equations of a curve are given by$x=e^{\sqrt{4t}}$and$y=\sqrt{e^{6t}}$. My attempt, I found$\frac{dx}{dt}=\frac{e^{2\sqrt{t}}}{\sqrt{t}}... 1answer 26 views ### Parametric Equations (Concavity) The question is: A curve is defined by the parametric equations $$x = t^2 + a$$ $$y = t(t-a)^2$$ Find the range of values for t in terms of a where the function is concave up? What I have ... 0answers 45 views ### I don't understand these directional vectors I'm currently practicing for my final calculus exam in 20 days time, it has a vector section, here is my problem. When I need to find these parametric equations for either lines of planes, I need a ... 2answers 65 views ### Path of a cycloid In this question, it's said that the path of a cycloid can be given as this parametric equation: \begin{align*}x &= r(t - \sin t)\\ y &= r(1 - \cos t)\end{align*} and is shown here: ... 1answer 93 views ### Parameterization of a torus Given that the parameterization of a torus is given by:x(\theta,\phi) = (R + r\cos(\theta))\cos(\phi)y(\theta,\phi) = (R + r\cos(\theta))\sin(\phi)z(\theta,\phi) = r\sin(\theta)$and the ... 0answers 38 views ### How do I detect collisions in a parametric equation$?$(Lissajous curve)$?$Let's say you have a simple parametric equation where$x= \sin(3t)$and$y=\cos(7t).\$ This is a pretty simple parametric equation that generates a relatively complicated Lissajous figure. You can ...
How do I prove that a hypocycloid, which has equation $$x^{2/3} + y^{2/3} = a^{2/3}$$ can be parameterized by $$x = a\cos^3(\theta),\qquad y = a\sin^3(\theta)$$? The problem assumes that it is true, ...