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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.)

2565 questions
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2 lines passing Q and R meets at the mid-point,

Consider the straight line whose parametric equation is $$(x, y) = (1, 1)+ t(12,−1)$$ Show that the above line and a line passing Q and R meets at the mid-point. $Q = (5, 5)$ and $R = (9,−4)$ How do I approach this problem? Any help and direction…
almost a beginner
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In parametric equations , how can the resulting equation after eliminating $T$ , consist of points not on the original set of equations?

I'm doing my Math level $2$ SAT subject test and there is a problem in the book that says "The resulting equation of eliminating $t$ may consist of points not on the graph of the original set of equations" So, can you explain this for me with maybe…
Marah
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Parametric equation with image of a function.

Find all values of $a$ for which the image of the function $$y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1}$$ contains $[2, 3]$. Now, I've already transformed it to $$y=\frac{(\sqrt{a}+1)-2\cos x}{(\sqrt{a}+1)^2+1-\cos^2 x}$$ And in…
mniip
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Parametrization by arclength

I could not re-parametrize the curve r[s_] := {-(5 + 2*Cos[2*s])*Sin[3*s], (5 + 2*Cos[2*s])*Cos[3*s], 2*Sin[2*s]} neither by hand nor with Mathematica. Is there any method else to parametrize the curve with uniform velocity (edit: not velocity,…
user230541
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Finding points on a tangent line of a parametric equation that are parallel to another parametric equation

So I got the tangent line of the first equation to be 12t/(3t^2+4) and I changed the second parametric equation to the cartesian form and got y= -(12/7x+5) with 12/7 as my slope. I equated 12t/(3t^2+4)= 12/7 and solve for t using the quadratic…
user226550
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Parametrics Arc Length

Can anyone explain the difference between the arc length and total distance? I'm using the textbook here and they seem to be the same formula. Please help me figure this out.
Malik Brahimi
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Smallest interval for graph

Parametric equation of a graph is x = cos(4t) , y = sin(6t) What is the length of the smallest interval $I$ such that the graph of these equations for all $t\in I$ produces the entire graph $\mathcal{G}$? What is the process of finding smallest…
user3138594
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finding parametric equations from a rectangular equation

Find the parametric equations for $x^2-4x+y^2-2y+5=2$, and graph. Hint: Complete some squares. I have completed squares and gotten $(x-2)^2+4=-(y+1)^2-2$ but I am confused with how to proceed. I know it will be a circle but how do I change this into…
Brynn
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Reparametrization of angles

Why is it true that $(\cos (a \theta +b), \sin (a \theta +b), (c \theta +d))$ for $\theta \in [\theta_1,\theta_2]$ can always be written as $(\cos \alpha \theta' , \sin \alpha \theta', \beta \theta')$ for a suitable choice of $\theta'\in…
Mister
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Detect "cusp" in parametric curve

I'm using the word "cusp" informally here, I apologize if there is a formal definition for it. What I'm looking for is a point where the derivative is non-continuous, I think. I have a a sequence of two-dimensional points on a parametric curve…
brianmearns
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How do solve this equation with dot product?

$$[(2,-7)+t(2,10)]\cdot n_1=0$$ $$\text{Solving for $t$, we find }t=\dfrac5{12}$$ $n_1=(1,1)$ or $(-1,-1)$. How does $t=\dfrac5{12}$?
terry
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Convert vector parametric equation to general form

Given the equation of a plane $x$ is $$x(s,t)=(0,1,1)+s(1,0,1)+t(2,1,-1)$$ How can I convert this equation into the general form $$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$$ Thank you.
Jacky
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Write an equation for the line through $A =(3, 1)$ and $B = (1, 2)$.

The line passes through $B$ and is parallel to $B - A$. So, the equation is $X = B + t(B - A)$. My question is: can we say that the following equations are correct as well? $X = B + t(A - B).$ $X = A + t(B - A).$ $X = A + t(A - B).$
someotherperson
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What does $v = v_0 + t_1(v_1 - v_0) + t_2(v_2 - v_0)$ parameterize where $t_i$s are scalars and $v_i$s are vectors?

On the one hand $v$ looks like it describes a plane. On the other hand, $v_0 + t_1(v_1 — v_0)$ describes a line in $3$-space. Since we need two vectors(?) to describe a line, $t_2(v_2 - v_0)$ is redundant, meaning $v$ describes a line. Can you…
godlessliburul
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What does $v = v_0 + t_1v_1 + t_2v_2$ parameterize?

Let $v_1$ and $v_2$ be given vectors. $v = t_1v_1 + t_2v_2$ varies over the plane determined by the two vectors. The plane is parameterized by $t_1$ and $t_2$. Let $v_0$ be another given vector. What does $v = v_0 + t_1v_1 + t_2v_2$…
godlessliburul
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