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

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1answer
58 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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2answers
32 views

Find point(s) of intersection between a line and a circle whose radius is parameterized by the same variable as the line

Let's assume we have a line: $$\begin{align} x&: x_0 + v_xt, \\ y&: y_0 + v_yt \end{align}$$ and a circle $$\begin{align} x&: X_0 + kt\cos(s), \\ y&: Y_0 + kt\sin(s).\end{align}$$ ...
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3answers
39 views

How a formula is developed

The rule for converting line equations to parametric equations is: $$\frac{(x-x_1)}{a} =\frac{(y-y_1)}{b} =t$$ I would like to know how this was developed. Thank you.
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1answer
43 views

Find the end points of a line segment in 3D space

I have a line segment in 3 dimensional space (x,y,z), and I want to find the 2 endpoints of this line segment. Is there a systematic way of doing this? To be specific, I have the line described by ...
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1answer
48 views

Finding twice-differentiable function of x of a parametric curve when dx/dt = 0

We're working on finding tangents of parametric curves and I feel like this problem isn't as hard as I'm making it out to be, but I am completely stumped. I am given this information: Given $x=3cos(...
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1answer
44 views

Finding an equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$

How to find a parametric equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$, and while C be the the Boundary of that surface. While reading the solution of ...
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0answers
22 views

Find intersection between conotur point list and a line

Given: List of points representing a closed contour Task: Choose a random point on the contour and shoot a ray inside the contour and determine where the ray intersects the contour. This needs to be ...
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1answer
382 views

Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$ how can I find a equation for the surface the curve lie? Also what ...
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0answers
63 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g.
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4answers
194 views

Parametric to implicit form of a curve

"Find the implicit form of the curve defined by parametric equations $x = t+1,y=\frac{1}{t^{2}}$" How can I clear $t$ to arrive at the implicit equation?
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1answer
25 views

Parametric equations of a line

"Find the parametric equations of a line that passes through point $(1,1,0)$, parallel to plane $2x+3y+z=7$ and perpendicular to the line $\frac{x-1}{-2}= \frac{y}{3}=-z-2$" I don't know where to ...
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1answer
275 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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1answer
268 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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1answer
68 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
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1answer
33 views

Slope of a Parametrized Curve

Say that we have the parametrized curve $x=e^{3t}, y=te^{-t}$. What would be the slope of this at the point $(1,0)$ and also on which points on the curve would the curve be horizontal? What I have ...
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2answers
47 views

Given two curves, find parametric curve

I am given two graphs x versus t and y versus t and I have to determine the parametric curve. The two graphs I am given: Parametric curve (that is the right answer): So the solutions say that: ...
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0answers
45 views

Is it possible to turn the parametric equation of a line in 3 dimensions into the general equation?

I Know it is impossible to do so since the parametric equation for a plane is the intersection of $2$ planes.For example: $x$ $=$ $\frac{-5}{4t}+\frac{1}{4}$; $y=\frac{3}{4t}+\frac{5}{4}$; $z=t$ ...
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1answer
93 views

Surface integral: Cone cut by a cylinder

Ok I've got this exercise from Apostol I'm trying to do: "The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral: $$\int\...
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1answer
46 views

Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
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1answer
254 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...
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1answer
39 views

Counting the integer soultions to this parametric inequality

hello I am looking for an efficient way, hopefully a formula or a somewhat tight upper bound, for the number of integer solutions to the following let $k$ be a fixed integer and $\lambda \ge 1$ and $...
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2answers
40 views

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior. my works I'm stuck in next step
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2answers
25 views

Parametric equation question showing minimum value of d^2

for the equation $d^2 = (1-a)^2t^2 + 18(1-a)t +117$ Show that when $a = 2$, the minimum value of $d^2$ is attained when $t=9$. I set $a=2$ to get $d^2 = t^2 - 18t + 117$ should i now just run it ...
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2answers
18 views

Rearranging this equation

This is based on a parametric equation problem. We have two ships A and B at $(-2,at +1)$ and $(4, t+10)$ respectively. I need to show that $d^2 = (1-a)^2t^2 +18(1-a) t +117$ using the distance ...
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2answers
161 views

What's the parametric equation for the plane through a point (x,y,z) perpendicular to (a,b,c)?

Find the parametric vector and Cartesian equations for the following planes: a. The plane thru point $(2,1,-2)$ perpendicular to vector $(-1,1,2)$. b. The plane thru the three points $(2,2,-2)$, $(-...
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2answers
58 views

Curl of a vector field.

Let S be a piecewise smooth oriented surface in $\mathbb{R}^3$ with positive oriented piecewise smooth boundary curve $\Gamma:=\partial S$ and $\Gamma : X=\gamma(t), t\in [a,b]$ a rectifiable ...
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2answers
22 views

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 ...
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3answers
291 views

Find the shortest distance between the point and a parabola

Find the shortest distance between the point $(p,0)$, where $p> 0$, and the parabola $y^2=4ax$, where $a>0$, in the different cases that arise according to the value of $p/a$. [You may wish to ...
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1answer
135 views

finding a vector valued function for the intersection of two shapes

I have a problem for my cal 3 class to find a vector valued function for the intersection of these two equations. $4x^2+4y^2+z^2=16$ and $x=z^2$ so i know that the first equation is a ellipsoid ...
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1answer
148 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
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1answer
43 views

The area of surface obtained by rotating a rectifiable curve

Let $\Gamma :X=\gamma(t),a\leq t\leq b$ be a rectifiable parameterized curve in the $(x,z)$-plane of $R^3$, which means $\gamma:[a,b]\to R^3$ is a $C^1$-mapping with $\gamma(t)=(x(t),0,z(t))^T$ and $\...
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1answer
118 views

How to parameterize a straight line?

Why does the straight line from $(x_1,+y_1,+z_1)$ to $(x_2,+y_2,+z_2)$ become $r(\vec t)=(1-t)(x_1,+y_1,+z_1)+t(x_2,+y_2,+z_2)$ for $0 \leq t \leq 1$?
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0answers
36 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
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1answer
20 views

Derivative of the magnitude of a parametric function

I am trying to show that $d/dt$ $|r(t)|^2 = r(t)*r'(t)$, where $r(t)= <x(t), y(t), z(t)>$ and $r(t) \neq 0$. I first tried using the fact that $|r(t)|^2 = (x(t))^2+(y(t))^2+(z(t))^2$ and then ...
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3answers
42 views

Parametrization of $x^2+y^2-ay=0$

I am to find the circulation of $$y^2 dx + x^2 dy$$ along the (counterclockwise) path $$\Gamma : x^2+y^2-ay = 0$$ both with and without using Green's theorem. Apparently, $\Gamma$ is supposed to ...
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2answers
115 views

Parametrization of an intersection cylinder ellipsoid

I'm trying to parametrize the surface given by the equations : $$\frac{x^2}{2}+\frac{y^2}{2}+z^2=1$$ and $x^2+y^2=y$. I found this function : $f:[0,1] \times [0,2\pi] \to \mathbb{R}^3$, $$(r,x) \...
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1answer
68 views

Find parametric line between two 2D line segments that is an exact distance from a point

Given two 2D line segments, $\overline{ab}$ and $\overline{cd}$, and a point $p$, I would like to find a scalar value $t$ such that the line segment between $\overline{ab}(t)$ and $\overline{cd}(t)$ ...
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1answer
29 views

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 ...
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2answers
69 views

Find the plot of $y=1+\cos t$, $x=\sin^2t$.

I'm trying to find the plot for the following : $$y=1+\cos t, x=\sin^2t$$ I'm trying to get ride off variable $t$. This is what I done for some reason is incorrect : $$x=\sin^2t=\frac{1}{2}-\frac{\...
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1answer
26 views

Finding the coordinate at time $t$ of a line determined by the points $(x1,y1), (x2,y2)$

I have the problem here, I create a program that clipping a line with the input (x1,y1,x2,y2). but the algorithm only explain until I get ...
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1answer
201 views

Evaluating the Surface Integral $\iint_{x^3+y^3+z^3=a^3} \frac{\bf{x}}{||\bf{x}||} \cdot d\bf{S}$

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}) \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
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2answers
50 views

Converting parametric equations with trigonometric functions into Cartesian form

Ahoy, I am having trouble with a computer-based assignment and the question is as follows: $$x = 2\cos^5 t, \quad y = 2 \sin^5 t$$ Write these in Cartesian form, $F(x,y) = c$. I understand ...
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3answers
55 views

Parametric Curves and Tangents

I am struggling with a question regard parametric curves and finding tangents to them but something is going wrong somewhere in the process and I cannot figure out why. The question asks: consider ...
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3answers
70 views

Consider the parametric curve given by: $x=3\cos(t)$, $y=t^{3/2}$.

The question asks to find the equation of the tangent to this curve at the point $t=\pi/4$. I've determined $$\frac{dy}{dx} =(\frac{dy}{dt})/(\frac{dx}{dt}) = -0.222$$ Have I got the right idea? ...
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3answers
57 views

Consider the parametric curve: $x=6\cos^3(t), y=6\sin^3(t)$, write it in cartesian form.

Consider the parametric curve: $$x=6\cos^3(t), y=6\sin^3(t)$$ Write it in Cartesian form. I am really struggling with the solution for this. I've been trying to find $t$ from $x$, and then plugging ...
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1answer
40 views

Parametric parabola

I was given my Math C assignment today and the moment I looked at question 1 I knew I had no idea what to do. This is the graph I was given: I was asked to provide an equation for the curve however ...
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1answer
77 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
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1answer
50 views

Looking for a family of astroids

I'm wondering what's the formula for a family of curves. Specifically the astroid. A few requirements: There should be one main one and then a bunch of them nestled inside. At each of the cusp-...
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2answers
41 views

Finding answers to system of equations

Let's say we have such a system structure of equations: ...
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1answer
160 views

Parametric equation of clock hands

I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to $...