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

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14
votes
6answers
9k 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 ...
14
votes
5answers
25k 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 ...
45
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) ...
26
votes
2answers
373 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 ...
10
votes
2answers
3k 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 ...
2
votes
6answers
506 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
2
votes
4answers
883 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
27 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 ...
1
vote
1answer
5k views

find parametric equations for the path a particle that moves along the circle $x^2+(y-1)^2=4$

Find parametric equations for the path a particle that moves along the circle $$x^2+(y-1)^2=4.$$ In the manner describe a) One around clockwise starting at $(2,1)$ b) Three times around ...
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
920 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 ...
11
votes
3answers
233 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 ...
6
votes
3answers
13k 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 $$ x_1 = 6-6t-6s$$ ...
2
votes
2answers
80 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, ...
2
votes
1answer
315 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 ...
6
votes
3answers
45k 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 ...
1
vote
1answer
992 views

Parametric Equations of an Oblique Circular Cone

I am trying to determine the parametric equations for a specific shape of an oblique circular cone with no success. Exhaustive web searchs and many texts have not been fruitful as regards ...
2
votes
1answer
62 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 ...
2
votes
2answers
4k 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 ...
1
vote
1answer
45 views

Show in parametric form plane in $R^3$

I have this question in my book. Show in parametric form the plane of $R^3$ that determined by these points : $$(1,0,0)$$ $$(0,1,0)$$ $$(0,0,1)$$ Does $(0,0,0)$ found on this plane? My answer The ...
1
vote
2answers
163 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
0answers
54 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 ...
0
votes
1answer
42 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 ...
0
votes
1answer
101 views

Parametrization for implicit function

$3y^2=x(1-x)^2$ By differentiation we can knwo that the sketch of this graph has one circle. I want to draw a graph in maple. Implicit plot does not work well So I will use parametric way. ...
0
votes
1answer
765 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 ...
0
votes
2answers
958 views

Explanation of the area under the curve given by a parametric equation

My textbook says the area under a graph is given by: $\smallint ydx$ And it then goes on to say by the chain rule: $$\smallint ydx = \smallint y{{dx} \over {dt}}dt$$ Could someone explain to me how ...
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
3answers
3k views

How to find a parametric equation for the tangent line to the curve of intersection of the cylinders?

How can i find a parametric equation for the tangent line to the curve of intersection of the cylinders $x^2 + y^2 = 4$ and and $x^2 + z^2 = 1$ at the point $P_0(1,\sqrt{3}, 0)$?
2
votes
1answer
263 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. ...
1
vote
3answers
69 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
3answers
51 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 ...
1
vote
1answer
548 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 ...
1
vote
2answers
176 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
2answers
101 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
1answer
85 views

Looking for help for building a Spline's algorithm 10th order

I'm trying to code the following algorithm in C++ and need help to understand the build of Splines from a mathematical point of view (found on page 129 on this paper). $$ f(t) = \boldsymbol{t} \cdot ...
1
vote
1answer
65 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
95 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
510 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
938 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
173 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
156 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: $$ ...
1
vote
4answers
848 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
601 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
2answers
674 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: ...
1
vote
1answer
358 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 ...
0
votes
2answers
57 views

Derivative with parametric set of equations

I am struggling for hours on a fairly simple problem. I have the following set of parametric equations: $$i = I (q_1^2 + q_1 - q_2^2 - q_2)$$ $$v = V (2(q_1 - 1) + \log q_1 - 2(q_2 - 1) - \log q_2)$$ ...
0
votes
2answers
59 views

Converting from Parametric to Cartesian?…

I've been working on converting from parametric equations into cartesian form, but can't figure out this problem? $$x=(t^2+1)/(t^2-1)$$ $$y=(2t)/(t^2-1)$$ How do I covert that to Cartesian? Any help ...
0
votes
1answer
18 views

What is the vector equation of the line through the head of $v_0$ and parallel to $v_p$?

$v_0$ and $v_p$ are vectors. Let $v_0, v_1$ and $v$ be vectors, all emanating from $(0, 0, 0)$. Suppose the line $l$ is passing through their heads. Let $v_p$ be on the line $l$ such that $v_1 = v_0 ...
0
votes
1answer
17 views

How can I parameterize these angles

My angle starts at 90, goes down to 0/360, then down to 270. This is a 180 degree range of motion. How can I express these angles from 0 - 180 instead of 90 - 270, where 90 gives 0 and 270 gives 180. ...
0
votes
2answers
2k 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)?