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

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24 views

Show that TU is perpendicular to the axis of the parabola.

The parabola is $x^2=4ay$ Show that TU is perpendicular to the axis of the parabola. Information given: Points of P$(2ap, ap^2)$, Q$(2aq,aq^2)$, and R $(2ar, ar^2)$ lie on the parabola ...
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2answers
44 views

How to set control points for spline curves

I've written a program that calculates points on spline curves (including Hermite, Bezier, and B-splines) and plot the curve on the screen (the curve is plotted on an html canvas using javascript). ...
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2answers
22 views

Parametric Equation of conics: Parabola

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$. a) Show ...
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13 views

Equations of an Oblique Circular Cone ($2$ circles are known)

I am trying to determine the parametric equations for an oblique circular cone with no success, as is shown in the figure above. Top circle is at point $(31,30,125)$ with a radius of $20$, and ...
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1answer
19 views

What is the purpose of $v$ in the parametric equation for a sphere?

The longitude / latitude parameterization of a sphere is described by: $x = cos(φ) * cos(θ) \quad y = cos(φ) * sin(θ) \quad z = sin(φ)\quad$ where $\quadθ = 2 π u$ and $φ = π v - π / 2$ I ...
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1answer
25 views

In which cases is there a need for U and V in parametric equations

I'm am reviewing parametric equations (to get a better grasp over how they are used to make shapes in computer graphics) and currently I have an understanding of how the parameter $t$ is used to ...
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17 views

Finding the intersection of two parametric equations? || How does one solve $x^x = n$? [duplicate]

Specifically: $c_1: x = t^t, y=t$ and $c_2: x= 81, y=t$ When trying to solve it, I'm coming up with: $y^y = 81$, and $y = y$ which is basically what I started when trying to solve $x^x = 81.$ ...
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2answers
32 views

Is there a way to parametrise general quadrics?

A general quadric is a surface of the form: $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$ It can be written as a matrix expression $$ [x, y, z, 1]\begin{bmatrix} A && ...
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1answer
19 views

Is my parametrisation correct?

Find a parametrisation $f\ : \ [a,b] \rightarrow \mathbb{C}$ for the line segment from i to 1 + i. So my answer is $f : [0,1] \rightarrow \mathbb{C}$ defined as follows f(t) = (1 - t)i + t(1 + i)
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3answers
31 views

parametric equation $x=2\ln (t+2)$and $y=t^3+2t+3$

Given that the parametric equation $$x=2\ln (t+2)$$ $$y=t^3+2t+3$$ At the point P on the curve, the value of the parameter is p. It is given that the gradient of the curve at P is $\frac{1}{2}$. Show ...
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1answer
29 views

Is this parametric equation describe a circle?

Let $w=\varepsilon\beta(t)-i\sqrt{\beta(t)^2-1}$, where $\beta(t)=\cosh t$ and $\varepsilon >0$. the parametric function is defined as $x+iy=\frac{2w}{|w|^2+1}$ and $z=\frac{|w|^2-1}{|w|^2+1}$. ...
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2answers
35 views

Parametric curve: $x=\frac{a}{2}(t+\frac{1}{t})$, $y=\frac{b}{2}(t-\frac{1}{t})$?

What kind of shape is the parametric curve described by: $$x=\frac{a}{2}(t+\frac{1}{t})$$ $$y=\frac{b}{2}(t-\frac{1}{t})$$ $a,b \in\mathbb{R^+}$ ?
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2answers
51 views

On my grapher, $(\cos t, \sin (t+1))$ generates a geometric figure. What is that figure?

On Wolfram Alpha I am getting a graph like this: http://www.wolframalpha.com/input/?i=%28sin+t%2Ccos%28t%2B1%29%29 Is this an ellipse? I really don't know how to find it.
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18 views

Existence of vertical or horiaontal tangent vector of a circle with different representation of the same curve

I am recently working with this question but not sure if I am on the right track, which is: $$t\mapsto (\frac12 cos(t^2), \frac12 sin(t^2)),0\le t \le \sqrt{\pi} $$ which clearly is a semi-circle ...
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2answers
41 views

Is there a general equation for an n-ellipse?

I'm sorry if this question is too trivial, but even a more thorough search on Google brought me no answers so far. So please, is there a general equation for n-ellipses? Given N points on the ...
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2answers
166 views

Parameterization of an ellipse

If an object (like a planet) orbits around a more massive object (like the sun) the orbit will be an ellipse with the massive object at one of the two foci of the ellipse. The parameterization $$x(t) ...
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1answer
16 views

Parametrics, when $t$ is not in between $0<t<1$

I understand how to parameterize a line segment when the $t$ value lies in between $0$ and $1$, however I was wondering how to create a parametric equation for the line segment between say $(1.5,2)$ ...
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3answers
80 views

Trigonometric equation with parameter

Find $p$ for which $\cos^2(x) - \cos(x) + p + 1 = 0$ has EXACTLY two solutions for $0 \le x \le 2\pi$ I tried to substitute $t = \cos(x)$ and then I got two solutions, but I don't know what to do ...
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1answer
21 views

Area of parametric surface (theory)

In the picture below $\left \|\Delta u_i r_u \times \Delta v_i r_v \right \|$ is the area of the parallelogram $\Delta T_i$ Can someone please explain why the sides of the parallelogram $\Delta T_i$ ...
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2answers
48 views

Finding an equation relating $x$ and $y$ with their respective parametric equations and using its differential?

How can I find the equation relating $x$ and $y$ directly without an additional parameter $t$, which both are related to initially. For example, $\frac{\mathrm{d}y}{\mathrm{d}t} = 2t+1$, ...
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1answer
27 views

Parametric line segment in 3-space

If one wants to parametrize a straight line segment in $\mathbb{R}^3$, which goes from $(1,0,0)$ to $(0,1,\pi/2)$, would this approach be correct? First, we come up with the $xy$-plane equation, ...
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3answers
48 views

Turning points of parametric curve

Find the slope of the curve at $t=\frac{1}{4}\pi$. $$\begin{cases}x=\sin t+\cos t \\ y=\frac{1}{2}\sin(2t)\end{cases}$$ $$\frac{dy}{dx}=\frac{\cos(2t)}{-\sin t+\cos t}$$ ...
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1answer
33 views

Turn the direction of movement of a parametric curve

Given is the parametric curve $K$ that satisfies $$\begin{cases}x=3\sin t \\ y=2\cos\left(t-\frac{1}{4}\pi\right)\end{cases}$$ How can you change the parametric equations if you want to turn the ...
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3answers
23 views

Parametric differentiation

A question from Active maths. At 60 years old this is my interest not my homework!! Let $$\begin{cases}y=e^t\cos t\\ x=e^t\sin t,\end{cases}$$ and prove that $dy/dx=\tan(\pi/4 -t)$. I ...
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1answer
45 views

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 ...
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1answer
17 views

Numerically Invert the Wakeby Percent Point Function

I am looking at the possibility of using the Wakeby Distribution to attempt to model color components in image rows and columns (it is a very silly idea for "compressing" images that I want to see ...
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1answer
51 views

Arclength of parametric curve

Find the arclength of the curve defined by $$x=\cos^2(t)$$$$y=\cos(t)$$ from $0$ to $4\pi$. I know using the formula that the arclength is given by ...
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2answers
58 views

parametric equations of a curve

The parametric equations of a curve are $$x=2\theta-\sin 2\theta$$ $$y=2-\cos 2\theta$$ The question asks that ''For the part of the curve where $0<\theta<2\pi$, find the coordinates of the ...
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0answers
40 views

Mathematic's difficulties to understand the parametrisation of an electrostatic potential field

I start to learn electrostatic and I have some problem with finding the upper and lower boundaries of my parametric variable that I used to represent the graph of the potential surfaces of two ...
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2answers
47 views

parametric equations with cubed sin and cos

It has been a while since I have had calc 3, I know how to find the rectangular equation from parametric equations; however, I do not remember how to find the rectangular equation given these ...
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1answer
28 views

Parametric differentiation for equation of a tangent.

Given $y=t^3-\frac{5}{2}t^2$ and $x=\sqrt t$, for $t>0$, a) Use parametric differentiation to express $\frac{dy}{dx}$ in terms of $t$ in simplified form. b) Show that $\frac{d^2y}{dx^2}=at^2+bt$, ...
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1answer
38 views

Unparametrize $x = 7 cos t, y = 4 tan t$

"Express the given parametrization in the form $y = f(x)$ by eliminating the parameter. $x = 7 \cos t, y = 4\tan t$" $y=\pm4 \sqrt {\frac {49} {x^2} - 1}$ Is correct?
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1answer
73 views

Maximize velocity with parametric equations

Suppose we are asked to find the value of t at which an object is at its maximum velocity, if it travels on a path governed by: $x = 2 + 8cos(t)$ $y = sin(t)$ Here's what I understand: ...
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73 views

How to verify a plane is parallel to a line?

Plane equation is: $$7x-5y+2z-9=0$$ Line parametric form: $x=t, y=-4+3t, z=9-4t$
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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: $$i = I (q_1^2 + q_1 - q_2^2 - q_2)$$ $$v = V (2(q_1 - 1) + \log q_1 - 2(q_2 - 1) - \log q_2)$$ ...
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2answers
26 views

Parametric equations - stating values.

A curve is defined by the parametric equations: $x=\cos2t, y=\sin2t, 0<t<π.$ a) Use parametric equations to find $\frac{dy}{dx}$. Hence find the equation of the tangent when $t=\frac{π}{8}$. ...
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1answer
17 views

General vector curve for the surface of a sphere

I want to prove a theorem about a vector curve ${\bf c}(t): \mathbb{R} \to \mathbb{R}^3$ (for $t \in [a, b]$) which lies on the surface of a sphere in $\mathbb{R}^3$. It is in my understanding that ...
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1answer
76 views

Find a vector function represented by the curve of intersection?

I'm struggling with the following problem: Given $\, z = \sqrt{x^2 + y^2}\,$ and $\, z = y+1\,$ find the vector function represented by the curve of intersection of the surfaces using the ...
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1answer
56 views

solving for initial velocity using the position vector

I am having trouble wrapping my head around this problem. The big picture is that i have to calculate the initial velocity v= needed for a soccer ball to cross a goal line. this is a homework ...
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1answer
26 views

Constructing parametric equation for $x=3z\cos(\ln z)$

I was trying to transform this $$x=3z\cos(\ln z)$$ in parametric form: $$x=x(t)$$ $$z=z(t).$$ To this end I made a substitution $\ln{z}=t$ and I got: $$x=3{e^t}\cos(t)$$ $$z=e^t$$ $$t\in ...
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1answer
48 views

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, ...
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2answers
538 views

Parameterization of a curve for complex integral

I have problem with parameterization of a curve in order to evaluate a complex integral. Most docs that I've tried to read didn't explain the topic very well, especially, in case where the curve in ...
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1answer
92 views

Finding the parametric equation for a longbow curve about a circle

In the figure the circle of radius $a$ is stationary, and for every $\theta$, the point $P$ is the midpoint of the segment $QR$. The curve traced out by $P$ for $0<\theta<\pi$ is called the ...
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1answer
41 views

How to solve parametrized limits?

I am a bit confused at the moment. This exercise in particoular shattered my self confidence: $$\lim\limits_{x\to 0}\frac{\sinh(x) + 1 - (1 + 3x)^{\frac13}}{\ln\left(1 - 2x^α\right) + 2x^3}$$ with ...
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1answer
64 views

What's the equation of this parametric surface?

disclaimer: my math is sketchy at best AND english is not my first language, so... i might have some issues naming things - but i'll try my best to be clear :) given this parametric curve: see it ...
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45 views

finding t for parametric equation tangent line

Here is the problem I'm trying to solve: Find the tangent line at the point (0,2) $$x=2 \, \cot(t)$$ $$y=2 \, \sin^2(t)$$ $$\frac{dy}{dx} = -2 \, > \sin^3(t) \, \cos(t)$$ The tangent line ...
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1answer
16 views

Proof: Given a $C^1$ parametrization of $f=f(x,y):\mathbb R^2\to \mathbb R$, substitute variables $(x,y) = (s+t,s-t)$

I'm studying Mathematical Analysis and trying to solve example problems as I go. Specifically, this problem comes after an introduction to $C$ parameterizations of $r:I\subseteq\mathbb R \to \mathbb ...
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1answer
61 views

On the implicit function theorem and the gradient.

I was following some MIT notes and came across this proof I had a doubt about the existence of $r(t) = \langle x(t), y(t), z(t) \rangle$ a parametrization of a curve on the level surface. Then I ...
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2answers
39 views

Need help to understand a math task about algebraic and parametric equations

Can anybody please explain this for me?: Find the algebraic and parametric equations of the circle with centre (-2,3) that passes through (1,-1) How do I find the algebraic and parametric ...
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1answer
32 views

Parametric curve parametriced by length

Normally you have a parametric curve with a variable t and you increment t to find the point along the curve. Is it possible to have a curve so that given a value it will give you the point on that ...