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Questions tagged [parametric]

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

In mathematics, a parametric equation of a curve is a representation of this curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter. This contrasts with implicit equations that define a curve as the zero set of some equation in the coordinates.

The parametric forms of curves are well-suited for drawing on a computer, while their corresponding implicit forms are useful for analytic manipulations (intersections, etc.)

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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 counterclockwise starting at $(2,1)$ c) halfway around…
user32104
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Show that the parametric equation $ x=x_1+(x_2-x_1)t , y=y_1+(y_2-y_1)t$

Can anyone help me to solve this? Show that the parametric equation $ x=x_1+(x_2-x_1)t $ $ y=y_1+(y_2-y_1)t\ $ with $(0\le t\le 1)$ describe the segment that joint the point $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ Thanks all
user32104
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How to define a finite objects with parametric equations

I never had seen parametric equations, but while trying to learn line integrals through Wikipedia, quickly found these equations are remarkable. Some can represent things for which more normal equations or functions are needed, if at all possible.…
JMCF125
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Can i formulate any curve with parametric equation?

Can i formulate any curve with parametric equation ? if not, so what kind of curves can be explained with parametric equations ?! Thanks in advance
farzad
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Parametrization of Circle in 3D

I'm given the vector valued function (supposedly a circle) $r(t) = (3\cos t, 4\cos t, 5\sin t)$. However, I cannot see immediately how this is a circle. How do I verify that it is? I also have a related question: how do I find the parametric…
John Daley
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Parametric equations for 3D surface

I have an equation x² + y² + z³ - z² = 0 This is an alpha-loop rotated around the z-axis. Can someone please help me to convert this to three parametric equations for x, y, and z (varying for u and v)? I ask because I am an artist trying to make…
Paul St George
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How to find a t value of a parametric given the X and Y coordinates?

(Sorry for the seemingly simple question) I need to find the t value of a set of parametric equations that corresponds to an (x, y) point on the parametric curve. The parametric curve will always be a circle. I'm given: The parametric equation will…
drobot
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Analyse the curve from parametric equations $x = \sin \theta, y = (1 + \sin \theta)\cos \theta$

A curve joining the points $(0,1)$ and $(0,-1)$ is represented parametrically by the equations $x = \sin \theta, y = (1 + \sin \theta)\cos \theta$, where $0\le\theta \le \pi$ Find $\frac{dy}{dx}$ in terms of $\theta$ and determine the x, y…
Steblo
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How do I prove that $x,y$ and $z$ lie on the same line if they are related to the same parameter?

Related to this question How exactly does that work? I would be grateful for a proof but barring that just the name of the theorem, so I can look it up myself. My work/thoughts so far: If we start with a system like the one…
Magnus
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Finding $a$ such that $ \sqrt{\frac32x^2-xy+\frac32y^2}=x\cos a+y\sin a$ has at least one solution other than $(0,0)$

Find all values of the parameter $ a $ from the interval $ [0, 2 \pi) $, for which the equation $$ \sqrt{\dfrac{3}{2}x^2 - xy + \dfrac{3}{2}y^2} = x \cos a + y \sin a $$ has at least one solution $ (x, y) $ other than $ (0,0) $.
Vladislav Kharlamov
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Parametric to Implicit ( {x(t),y(t)} --> P(x,y) == 0 )

I have this parametric equations: $x(\theta) = r Cos(\theta) - \frac{v_{0}^2Cos(\theta)Sin(\theta)}{g}$ $y(\theta) = \frac{v_{0}^{2}Cos^2(\theta)}{2g} + r Sin(\theta)$ This is for $\theta \in (-\frac{\pi}{2},\frac{\pi}{2})$ The last part of the…
José D.
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Parametric equation representation of a parabola's portion

The question states: The portion of the parabola $y=x^2+7$ from $(-5,32)$ to $(4,23)$. Since the $x$-values increase, what I did was write $x=-5+t$. Then, I did $y=x^2+7=(-5+t)^2+7=t^2-10t+32$. I gave the $t$-values from $t=0$ to $t=9$. This ensures…
Ansar Al
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Parametrizing Equations

I have the equation $4x^2+y^2=4$. My instinct when parametrizing it was to do $2\cos(t)=x$ and $\sin(t)=y$ because the 4 is attached to the x. But, as you probably know it is really $x=\cos(t)$ and $y=2\sin(t)$, which is very clear from a graph. …
Burt
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Parametric equation of a line knowing the trajectory of the center of a ball rolling on it

I have a parametric curve (in polar coordinates) that describes the trajectory of the center of a rolling ball. This ball (assimilated as a circle) rolls smoothly along a relief. I need to get an expression for the curve that describes the…
Anne Aunyme
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Intersection of Plane Through Origin and Sphere

I have a plane $$x + y + z = 0$$ and a sphere $$x^2 + y^2 + z^2 = R^2$$. How do I find a parametric equation for the intersection of the plane and sphere? The closest I've gotten was $r = (\sin{t} + \cos{t}, \sin{t} + \cos{t}, \cos{t}) $, but it…
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