Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [parametric]

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

In mathematics, a parametric equation of a curve is a representation of this curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter. This contrasts with implicit equations that define a curve as the zero set of some equation in the coordinates.

The parametric forms of curves are well-suited for drawing on a computer, while their corresponding implicit forms are useful for analytic manipulations (intersections, etc.)

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Distance between two parametric lines

Compute the shortest distance between the following two parametric curves, $\vec{r_1(t)}$ = $\langle -1+2t, 4-t, 2\rangle$ $\vec{r_2(t)}$ = $\langle 3-2t, 5+t, -1+3t\rangle$ I think this can be done by simply minimizing the distance function and…
John
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Equation of sine wave around a spiral?

I am trying to figure out the equation of a specific curve. This is for a CAD model that I am trying to generate. Essentially, I am trying to model a curve that would essentially look like a cross section of corrugated sheet metal, similar to a…
Justin Brim
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Parametric Equation of a Internal Point Rolling on a Wheel

A wheel of radius $4$cm rolls along the x-axis with angular velocity $2$rad/s. Find parametric equations for the curve described by a point on a spoke 2cm from the centre of the wheel if it starts from the point $(0,2)$ at time $t=0$. Since you…
Simon
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Different parametric functions give same result

While trying to parametrise the intersection between $x^2+y^2=z^2$ and $z=\frac x2+2$, polar coordinates gave me $$r^2=4+2r\cos t+\frac14(r\cos t)^2$$ which returned two values of $r$: $$r=-\frac4{\cos t\pm2}$$ Using the form $\{r\cos t,r\sin…
user381650
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parametric equation of two equations of one line

Hi I would like to find the parametric equation of the line: $\ \left\{ x+y=2 , 3x+y+z = 5 \right\} $ I have tried to solve it by posing $z = t $ and it gives me $\ \left\{ x=-(t-3)/2 , y = (t+1)/2, z =t \right\} $ but I don't know if it is…
Pierre-Luc Bolduc
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Finding parametric equations from lines

I have a question that reads: Find a parametric equation of each of the following lines: A. $3 x_1 + 4 x_2 = 6$ D. the line through $A=(-2,1)$ parallel to $x = (1,4) + t(3,5)$ E. the line through $A=(-2,1)$ perpendicular to $x = (1,4) + t(3,5)$ I…
user6259845
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parameterization of helical torus

A Helix is parameterized as $\langle R \cos(t), R \sin(t), \alpha t\rangle$ and one can visualize it as "wrapping" around a cylinder of radius R. I would like to accomplish the same thing but wrapping around a torus(or one can think of bending the…
Jubao
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$Q_{30}$ Graph the curves $y = x^3-4x$ and $x=y^3-4y$ and find their points of intersection correct to one decimal place.

I could not find all the points of intersection Here is what I did. First, I use elimination to obtain $(x_1,y_1)$, and $(x_2,y_2)$. For $y=x^3-4x$, I set $x_1 = t$, then $y_1=t^3-4t$. For $x=y^3-4y$, I set $y_2 = t$, then $x_2=t^3-4t$. Then I…
DSL
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Hi, I have been trying to understand the derivation of a hypocycloid's parametric equation, but am stuck with one part.

I have been using someone else's answer on the same site to understand the problem: here's the link - Parametric equations for hypocycloid and epicycloid I can understand everything but the part where it says - "The coordinate of the moving point…
paul
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Heuristics for putting $f(x_0,x_1,\ldots,x_n)=0$ into parametric form?

Suppose I have an implicit equation: \begin{equation} f(x_0,x_1,\ldots,x_n)=0 \end{equation} Which might be 'paramaterizable'; i.e. put it into the form: \begin{align} x_0 &= g_0(t_0,t_1,\ldots,t_m) \\ x_1 &= g_1(t_0,t_1,\ldots,t_m) \\ \vdots \\ x_n…
NauticalMile
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Collinearity of three points on a curve.

In the realm of elliptic curves, the collinearity of three points is of a fundamental importance because this condition allows us to define on the curve a law of Abelian group, the study of which is the subject of endless enigmas and guesses,…
Piquito
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Prove that $p^2+pq+2=0$

The information given is that a point $P(2ap,ap^2)$ on the parabola $x^2=4ay$. The normal to the parabola at P intersects the parabola again at $Q(2aq,aq^2)$. O is the origin of the graph. The equation of $PQ$ is $x+py-2ap-ap^3=0$ and the line PQ is…
Wilson Huynh
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Convert advanced parametric equation to regular/cartesian

can anybody help me to convert following parametric equation in a form Y =Y(X): $$ x = cos(t) \sqrt{(2 - cos^2(3t))} \\ y = sin(t) \sqrt{(2 - cos^2(3t))} $$ I've tried also with Wolfram Alpha and it seem not to work: Reduce[x == Cos[t] Sqrt[2 -…
Vito
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two-dimensional bounded area defined parametrically

How do I define this without using piecewise function? I think it has something to do with Bilinear Surface but not sure how to get started. $x_1=-1, x_2=1, x_3=0, x_4=1$ $y_1=0, y_2=1, y_3=1, y_4=0$ $…
Gavin
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How can I know whether the airplanes collide by using parametric equations

Recall that a line hes equation y=mx+c. Suppose one airplane moves along the line y=2x+3 while the other airplane moves along the line y=3x-2. By plotting a graph, even though the lines are intersect, but the equations did not tell us whether there…
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