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

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17
votes
6answers
11k views

Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated (...
16
votes
5answers
29k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
47
votes
5answers
4k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) &...
6
votes
3answers
53k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
28
votes
2answers
456 views

Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two ...
23
votes
2answers
828 views

Seeking proof for the formula relating Pi with its convergents

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via $\mathrm{A002485}(n)/\mathrm{A002486}(n)$ ?...
6
votes
3answers
16k views

Parametric form of a plane

Can you please explain to me how to get from a nonparametric equation of a plane like this: $$ x_1−2x_2+3x_3=6$$ to a parametric one. In this case the result is supposed to be $$ x_1 = 6-6t-6s$$ $$...
10
votes
2answers
4k views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a math-...
4
votes
3answers
5k views

Parametrisation of the surface a torus

For a calculus question I need to parameterise the surface of the torus generated by rotating the circle given by $(x-b)^2+z^2=a^2$ around the $z$-axis (with $0<a<b$). I've had a go at this, ...
2
votes
4answers
913 views

Sketch a curve given parametrically by $x=2t-4t^3$ and $t^2-3t^4$

I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too. $$x=2t-4t^3$$ $$y=t^2-3t^4$$ I can guess that the curve is a polynomial equation so in principle I can write this as $$w_1 ...
0
votes
0answers
48 views

Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function.

NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ...
2
votes
6answers
563 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
1
vote
2answers
47 views

Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$

I need to find and sketch the image of the straight line $z = (1+ia)t +aib$, where $-\infty < t < + \infty$, $a,b\in \mathbb{R}$, and $a \neq 0$, under the map $w = e^{z}$. In order to ...
1
vote
2answers
2k views

find area of the region $x=a\cos^3\theta$ $y=a\sin^3\theta$

Find the area of the region enclosed by $x=a\cos^3\theta$ and $y=a\sin^3\theta$ What steps should I take in order to find the area?
1
vote
1answer
623 views

How to construct a parametric cubic B spline?

If I am given n+1 control point Pi(xi,yi), Po .... Pn , how do I construct a parametric relationship to draw a curve ? From what I understand , a parametric relationship is that you can express x and ...
1
vote
1answer
1k views

Parametric Equation

Let $P_1$ be the plane through the origin containing the vectors $[1,2,-1]$ and $[0,1,1]$. Let $P_2$ be the plane through the point $(1,1,1)$ parallel to the vectors $[-1,2,2]$ and $[3,4,-2]$ I know ...
0
votes
1answer
29 views

Parametric Equations Problem

Im back! Um, i have a simple question im trying to get ready for test after 5 days.. I slacked of sadly :( on math, so i have to pick up my skills.. On my test review i have this question: The ...
11
votes
3answers
254 views

What function could describe this GIF animation?

I found this image on Beautiful Mathematical GIFs Will Mesmerize You and this GIF really caught my attention. From what I see, it's a 2D circle morphing into the 3D sphere. What function could ...
3
votes
0answers
43 views

Can parametric equations graph all kinds of lines?

I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example: ...
2
votes
1answer
375 views

Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then (...
2
votes
2answers
86 views

On the complete solution to $x^2+y^2=z^k$ for odd $k$?

While trying to answer this question, I was looking at a computer output of solutions to $x^2+y^2 = z^k$ for odd $k$ and noticed certain patterns. For example, for $k=5$ we have $x,y,z$, $$10, 55, 5\\...
2
votes
2answers
759 views

Parametric equations and specifications of a triskelion (triple spiral)

I haven't been able to find the parametric equations and specifications to form a triskelion, a triple spiral (this is made of three interlocked couples of spirals). Using the parametric equation of ...
2
votes
2answers
5k views

Derive parametric equations for sphere

How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\ z & = r \cos(\varphi), \end{align} where $...
2
votes
1answer
78 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 ...
1
vote
0answers
55 views

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where $a$...
1
vote
2answers
183 views

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ...
1
vote
2answers
3k views

Conditions for a smooth parametric curve

A curve defined by $x=f(t), y=g(t)$ is smooth if $f'(x)$ and $g'(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
0
votes
1answer
43 views

a question about how to parametrize a surface in $R^3$

Given a surface $$x^4/a^4+y^4/b^4+z^4/c^4=1$$,how can I parametrize the surface using X(u,v). I tried to use $x=a\sqrt{cos(\theta)sin(\phi)}$,$y=b\sqrt{cos(\theta)sin(\phi)}$,and $z=c\sqrt{sin(\phi)}$...
0
votes
1answer
878 views

Intersection of cubic bezier curve and circle

Let $B$ be a cubic Bézier curve with control points $P_0,P_1,P_2,P_3 \in \mathbb{R}^2$, and $C$ be a circle with center $P_C$ and radius $r$. How can I find all intersections of $B$ and $C$? Is ...
2
votes
2answers
790 views

Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation?

I have the following parametric equation: $$x=t^2-2t$$ $$y=\sqrt{t}$$ I'm interested finding the area of the region bounded by this curve and the y-axis (i.e. $0 \leq t \leq 2$). We have: $$\frac{\...
2
votes
2answers
78 views

Parametric equations where sin(t) and cos(t) must be rational

Suppose there are parametric equations $$ x(t) = at - h\sin(t) $$ $$ y(t) = a - h\cos(t) $$ and it is required that both $\sin(t)$ and $\cos(t)$ should be rational. What the values of $t$ should be ...
2
votes
1answer
289 views

Finding Angle of Elevation to hit X, Y

My ultimate goal is to find the angle of elevation necessary to launch a projectile from the origin to (x,y) with initial velocity V and under gravitational acceleration g. Wind resistance is ignored. ...
2
votes
2answers
51 views

Is parametric form of a given function unique? [closed]

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form ...
2
votes
2answers
182 views

Finding arc length parametrization of a parabola

Suppose we have a parabola of equation $y = x^2$ in a given Cartesian coordinate system. An obvious parameterization of it is the system $x = t$, $y = t^2$, but there are infinite other possibilities, ...
1
vote
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 ...
1
vote
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? ...
1
vote
1answer
1k views

Coordinate of intersection between line and square

TL;DR given a square and a point $p$, I need the intersection between the perimeter of the square and a ray cast from the center of the square through point $p$. This is my approach so far, but I will ...
1
vote
1answer
181 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
1
vote
1answer
107 views

What is being represented by this 2 images?

image 1 image 2 It's possible that image 1 is showing some kind of methods for building polygons out of trigonometric functions ? It's also possible that image 2 is a quadratic bezier curve ?
1
vote
1answer
368 views

Solving Parametric Equation: Multiple coefficients of trigonomic functions

How can I solve: $ x = 16 \sin^3(t) \\ y = 13\cos(t) - 5\cos(2t) - 2\cos(3t) - \cos(4t) $ I've derived $t = arcsin(\frac{x^\frac{1}{3}}{16^\frac{1}{3}})$ from the first equation but I am still unsure ...
1
vote
1answer
171 views

Find the Frenet frame

Consider the following space curve: $$ \gamma(x)=(e^x\cos(x), e^x\sin(x), e^x). $$ My main goal is to find the Frenet Frame T,N,B. So far I have found the arc-length using the following formula: $$ s(...
1
vote
1answer
170 views

How can I re-write an equation (or system of equations) in parametric form?

For the equation $y = 3x$ I need to re-write $x$ and $y$ in terms of a variable $t$. How can I find the value of each variable in terms of $t$?
1
vote
5answers
937 views

How do we prove that two parametric equations are drawing the same thing?

For example, if I have $$\begin {align} x(t) &= r\sin t\cos t\\ y(t) &= r\sin^2 t\\ \end {align}$$ and $$\begin {align} x(t) &= \frac r 2 \cos t\\ y(t) &= \frac r 2 (\sin t + 1) \...
1
vote
1answer
637 views

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 ...
1
vote
1answer
2k views

Finding Parametric Equations For A Rectangular Equation

I am trying to find a general way of finding parametric equations for a rectangular equation. The problem I am working on is $y=x^3$, and I have to find two examples of parametric equations. Obviously,...
1
vote
1answer
42 views

Parametric form of curves?

Can someone tell me the steps to get the parametric form of a curve? For example: $x^{2\over 3}$ +$y^{2\over 3}$ =1
1
vote
2answers
107 views

Describe a twisted parabolic trough

I want to describe a parabolic trough of the form $z=x^2$ and give it a twist, like a torsion in $y$ direction. Does anybody know how I can do that? Imagine this is the trough and the $z$ direction ...
1
vote
2answers
261 views

Catenary equation in 3D

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is lowest point of the catenary curve. I only know z-coordinate of this third point. I need to find ...
1
vote
1answer
66 views

Let $S$ be the surface generated by the circles of radius $b$, find a parametric expression for $S$

Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that $((x'(s))^2+((y'(s))^2=b^2$ and let $S$ be the surface generated by the circles of ...
1
vote
1answer
728 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...