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

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25 views

Express the region $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ as the union of cubic curves

Let $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ I know the family of curves $\gamma(t)=x^2+tx^3$ belong to D, for $t\in [0,1]$. It is true that for every $(x,y)\in D$ there exist a unique ...
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1answer
19 views

$\sin\left(ax + \frac{\pi}{6}\right)$, find $a$ with given slope

I was given a function $$y = \sin\left(ax + \frac{\pi}{6}\right)$$ In the point $x = \frac{\pi}{12}$, the slope of the tangent line of that point is $\frac a2$. I need to find $a$ if it's given that ...
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0answers
36 views

Mapping of lines/circles in complex plane by linear transformation

The problem So for the line i tried to use the functon g(t)= = a + tv (where t is real and v paralel to the line going through a) and plug it into the given function and from there i got it into a ...
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2answers
148 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, ...
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1answer
33 views

Calculating line integral on an intersection of two surfaces

I need to calculate the **absolute value ** of the integral: $$ \oint_C (4z+2xy)dx + (x^2+z^2)dy+(2yz+x)dz $$ where $C$ is the intersection of the surfaces: $z=\sqrt{x^2+y^2 }, x^2+y^2 = 2y$ . Will ...
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3answers
44 views

How to use parametric equations for line integral?

The question is to find the work done in moving a particle in a force field: $$\overrightarrow F = 3x^2i+(2xy-y)j+3k$$ along the straight line from (0, 0, 0) to (2, 1, 3) So, work done $=\int ...
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0answers
42 views

How can i translate a parametric equation to cartesian

The parametric equations are : $$ x=16\sin^3(t)$$ and $$y=13\cos(t)-5\cos(2t)-2\cos(3t)-\cos(4t) $$ with $t$ from $-\pi$ to $+\pi$ so I'm new to this kind of equations and i really don't know how to ...
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1answer
33 views

Parametric equations - finding A

The curve with parametric equations x = a(t-2), y = at² + 2 (where a≠0), meets the y-axis at the point (0,5). (a) Find the value of the constant a. (b) Hence determine whether the curve meets the ...
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0answers
20 views

Vectorial function representation

I'm trying to solve the following excersice: Given the following equations, represent by a vector function: $y=x^2+1$ from $(3,10)$ to $(-1,2)$ $y=4-x$ with $x\in [-2,3]$ ${x^2\over{25}} + ...
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2answers
416 views

Find the angle between two planes using their normal vectors

The angle between two intersecting planes is defined to be the angle between their normal vectors. Find the angle between the planes $x – 2y + z = 0$ and $2x + 3y – 2z = 0$. Find the parametric ...
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1answer
44 views

Normal to a parametric curve: $x=2t+3$, $y=2/t$ [closed]

A curve is given by the parametric equations $x=2t+3$, $y=2/t$. Find the equation of the normal at the point on the curve where $t=2$. I honestly do not understand how to do this question.
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10 views

Why is the parameter (t-1) in this example?

Example 1 on this page: http://mathinsight.org/parametrized_curve_tangent_line_examples Why do they use $(t-1)$ in the last step ($l(t)=c(1)+(t−1)c′(t_0)$)? Why not just use $t$?
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2answers
23 views

Finding cartesian equation from Parametric equation.

A curve $C$ is defined by the equations $$x=\frac{1+t}{1-t}$$ $$y=\frac{1+t^2}{1-t^2}$$ where $t$ is a real parameter. I found the $\frac{dy}{dx}=\frac{2t}{(t+1)^2}$. How to prove that $C$ has ...
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1answer
39 views

Transforming a sawtooth into a sinus with one parameter

Can you help me in finding the analytical expression of a function $f_\alpha(\theta)$, with one parameter $\alpha=(0,1)$ by which one can continously transform a sawtooth curve into a sinus? With ...
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1answer
45 views

Find the equation of the locus of R if the chord $PQ$ passes through $(0,a)$

The parabola is $x^2=4ay$ Information given: Points of P$(2ap, ap^2)$, Q$(2aq,aq^2)$, and R $(2ar, ar^2)$ lie on the parabola $x^2=4ay$. The equation of the tangent at P is $y=px-ap^2$ The ...
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32 views

Finding the locus of a point R

The two points $P(2ap,ap^2)$ and $Q(2aq,aq^2)$ are on the parabola $x^2=4ay$ The equation of the tangent to $x^2=4ay$ at an arbitrary point $(2at,at^2)$ on the parabola is $y=tx-at^2$. The tangents ...
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1answer
49 views

Prove that $p^2+pq+2=0$

The information given is that a point $P(2ap,ap^2)$ on the parabola $x^2=4ay$. The normal to the parabola at P intersects the parabola again at $Q(2aq,aq^2)$. O is the origin of the graph. The ...
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0answers
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
40 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
21 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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0answers
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
24 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
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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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17 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
145 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
78 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
41 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
23 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
46 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
22 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
41 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
45 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
42 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
27 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
67 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: ...