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

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19 views

What is the Implicitization Problem

Let $V$ be a subset of $k^n$ given parametrically as $x_1 =g_1(t_1,...,t_m) ...x_n=g_n(t_1,...,t_m)$. If the $g_i$ are polynomials (or rational functions) in the variables $t_j$, then $V$ will be an ...
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1answer
20 views

Sketching Parametrizations - how to get something more understandable?

So I have some parametric functions (of one variable) I'm trying to sketch. Generally I can do so by "reverse parametrizing" where I take $x(t)$ and make $t$ a function of $x$ and then substituting ...
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1answer
38 views

How to parameterise the curve $ x^2 = 4y, 3x^3 = 8z$?

As per title, I'm unsure how to parameterise the given curve? Are there different methods? I'm unsure about parameterisation in general, I just tend to remember specific formulas.
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0answers
16 views

Bezier Surface evalution

So the problem I'm having at the moment, is a thinking problem. I can draw a bezier surface (parametric surface) with 16 control points and if I evaluate S(u, v) I get a coordinate in the 3D space. ...
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0answers
36 views

Solving a complex exponential / logarithmic equation

I've found this interesting equation on the web: $$p-1 = (1 - e^{\alpha-\beta t})^{t+1}$$ which has to be solved for t, considering that the parametes: $\alpha, \beta, p$ are defined correctly. ...
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1answer
17 views

Is there a parametric form for a degenerate conic section?

With parametric form I mean a parametrization like $(\cos{t}, \sin{t})$ for a circle. A conic section has such a parametrization, but suppose it degenerates in 2 lines (ranges of points), is there a ...
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1answer
71 views

Parametric Equation of sine wave helically wrapped around a cylinder

I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body (3D). The parametric equation below gets me close to what I'm looking for, but not quite since the ...
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2answers
39 views

Parametric equation of a curve in $R^3$

How to find the parametric equation of the curve in $R^3$, which is the intersection of the sphere of radius $a>0$ centred at the origin, and the plane $x+y+z=0$? I've tried to start looking for ...
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123 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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1answer
36 views

Prove the normal will be at constant distance form origin in this parametric function?

Given a function, $x = a(cos \theta + \theta \sin\theta])$, $y = a(sin\theta - \theta\ cos\theta)$, $a \in R$ Prove that the normal drawn on each point is at constant distance form the origin? If ...
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1answer
38 views

Prove the continuity and differentiability of parametric integration

$$F(\alpha )=\int_{0}^{+ \infty } \frac{\cos x}{1+(x+\alpha )^{2} } dx$$ Prove the function F is continuous and differentiable on the interval $[0, +\infty )$
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1answer
24 views

Parametrization of a paraboloid part

Find the parametric equation of the surface $S$, where $S$ is the part of the paraboloid $z=x^2 + y^2 + 1$ bounded by the plane $z=2x+3$ My attempt The OXY projection of $S$ is $x^2 + y^2 + 1 = 2x + ...
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0answers
17 views

Parametrics Question Help Please [duplicate]

Would anyone be able to verify the answer of (iv) being $y=-a$? I assume since -a is a constant, the x value is irrelevant Thanks
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1answer
33 views

Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $ x+y-z+u=1 $ $ 2x+u=2 $ $ z -u=0 $ I don't ...
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4answers
41 views

Showing that these two lines are parallel.

$$ \dfrac{x - 1}{2} = 2 - y = 5 - z \quad \text{and} \quad \dfrac{4 - x}{4} = \dfrac{3 + y}{2} = \dfrac{5 + z}{2}. $$ I was given this math problem as homework, and I have spent the past hour ...
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1answer
24 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
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2answers
25 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
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1answer
39 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. ...
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1answer
36 views

Surface described by parametric equations

If I've got the surface in $\mathbb{R}^3$ described by: $x(s,t)=s^2-t^2$, $y(s,t)=s+t$, $z(s,t)=s^2+3t$ for $(s,t)\in\mathbb{R}^2$, and I'm told this surface is the graph of a function $f(x,y)$, how ...
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1answer
18 views

using parametric equations to form a line equation from two points

hi in order to form a line equation from two points i have been told to do following and not to use any other ways. a(9,6) b(2,-1) x=9-7t y=6-7t cancel out the ts gives x-y=3 but when the signs ...
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3answers
81 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
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0answers
44 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 ...
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1answer
39 views

Calculating curvature of a curve on a the surface $x^2+y^2=1$. [closed]

Find a curve on the cylinder surface $x^2+y^2=1$ in $\mathbb R^3$ such that its curvature is equal to $\frac1{100}$ at each point of this curve. Does this easily generalize to different surfaces?
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2answers
70 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 ...
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2answers
19 views

Paramaterizing a path $C$ along a parabola $y=2x^2$

I am doing a line integral where the path $C$ is defined as the arc of the parabola $y=2x^2$ from the points $(-1,2)$ to $(2,8)$. Is there a "catch all" approach or method that can be applied here? ...
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1answer
34 views

What is the non-piecewise curve that resembles the following roller coaster track?

I want to create an animation about roller coaster. One track I want to use looks like the following figure. I am looking for the simplest non-piecewise parametric equation for both $x(t)$ and ...
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2answers
42 views

How to determine the coordinate of roller coaster's wheels?

I want to create an animation about roller coaster. For a simple track, for example, a circle, I can determine the position of the center of its wheel easily. However, for any parametric curve, I ...
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1answer
28 views

Help find the equation of two planes

I have the question Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L MY ATTEMPTED SOLUTION I ...
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1answer
47 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
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2answers
39 views

Graphing a Parametric Equation

I need to graph and show the work for this problem. The graph needs to include arrows on the curve to show the direction of motion and I need to label the t-values graphed. $$c(t)=(2+4t, 3+2t)$$ So ...
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3answers
62 views

Finding the speed of a particle (parametric math)

I have to find the speed (as a function of $t$) of a particle whose position at time $t$ seconds is represtented by $$c(t)=(\sin t+t, \cos t+t)$$ How would I go about finding the maximum speed? ...
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1answer
31 views

Find the length of the curve

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$ If I understand correctly, I need to take the derivative of the y part of that coordinate over the ...
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2answers
49 views

Find the parametric equation to the curve

Find the parametric equation for the curve. $$x^{2}+y^{2}=10$$ I haven't learned parametric equations fully yet, so I wanted to check with you guys and see if you can confirm if I'm doing this ...
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1answer
34 views

Express the parametric equation in form of y=f(x)

I need to express the parametric equation in the form of $y=f(x)$ by eliminating the parameter. I haven't learned how to do this yet, I've attempted to read a few pages though but they didn't help me ...
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3answers
33 views

Parameter “m” for which $P(x)=4(m+1)x^3+(m-3)x+1-m$ has a root with multiplicity two…

Can you please help me solve this parametric problem. So, we have to find all the values of real parameter $m$ for which the following equation has a solution with multiplicity ...
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1answer
28 views

Find a right angle triangle in with 3 vertices and one parameter

Given three coordinates, which could be $A=(7,3)$, $B=(2,4)$, $C=(k,-2)$ I want to find the values of $k$ that make a right angle diagram out of the three points. So I initially was thinking to find ...
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1answer
34 views

$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$ For which $a$ has only one solution…

For which values of real parameter $a$ the following equation has only one solution: $$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$$ Frankly I have no idea and I hope you'll give me some understandable hint ...
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0answers
76 views

Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
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1answer
26 views

How do I change parametric in t to cartesian when I can't re-arrage for t

I'm stuck looking at this parametric equation which I have to put in cartesian form $x=t^2+ \frac1t$, $y=t^2-\frac 1t$ Something to do with difference of two squares? I can't see how to ...
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2answers
26 views

Parametric inequation…

Supppose we have $a$ a real positive number that's not equal to $1$. Solve the following inequation: $$\log_a(x^2-3x)>\log_a(4x-x^2)$$ If it's known that $x=3.75$ is one solution of it.
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1answer
16 views

Find the values of parameter $a$ so that…

Determine all the values of real parameter $a$ so that the equation:$$(x-a)[log_4(x-5)-1]=0$$ admits a maximum number of real solutions. Thank you!
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2answers
18 views

Parametric Eqn / Differentiation

Parametric eqns of a curve are $x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$, where $t$ cannot be $0$. At point $P$ on curve, $t = 3$ and the tangent to curve at $P$ meets the $x$-axis at $Q$. The ...
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0answers
24 views

Find all the values of real parameter “n”…

Let $S$ be the set of real solutions for the following equation:$$\log_2(1-x-x^2)=n\log_{1-x-x^2}2+2$$ Determine all the values of real parameter $n$ for which $S\cap(0;{1\over2})\neq\emptyset$.
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0answers
42 views

Mass and density function Calculus II Problem

A thin metal plate lies over the portion of the cylindrical surface $y^2 + z^2 = 4$ for $z ≥ 0$ between $1 ≤ x ≤ 4$. The density of the plate is given by $f(x,y,z) = z$. How do I calculate the mass ...
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3answers
49 views

Find the real parameter so as the equation has no real solutions…

My question is: For which values of parameter $a\in \mathbb{R}$ the following equation $$25^x+(a-4) \,5^x-2a^2+a+3=0$$ has no real solutions? My idea is: First of all we should transform the ...
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0answers
23 views

when is it not possible to eliminate parameter t from parametric equations?

We can eliminate parameter t from a set of parametric equations to convert to a Cartesian equation. My book mentions that it is sometimes impossible to do that. What would be an example of not being ...
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1answer
37 views

Surface area generated by revolving about the y-axis

I have to find the surface area which is generated by revolving the curve about the y-axis found below: $$x=\frac{1}{2}(e^{y} + e^{-y}) \ ; 0<=y<=ln2 $$ I know how to solve the question, when ...
2
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1answer
95 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 ...
0
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1answer
77 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
2
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2answers
94 views

Curve on a basketball

The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of ...