0
votes
4answers
40 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 ...
0
votes
1answer
28 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 ...
0
votes
1answer
72 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
votes
3answers
166 views

Equation of a torus

First I am a newbie in maths so please forgive me if I am not as rigorous as you would like, but do not hesitate to correct me. I want to find the equation of a torus (I mean the process, not just ...
3
votes
1answer
53 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
0
votes
1answer
50 views

Equation of a polygon

I need a parametric equation for a filled polygon defined by 3 or more points. The closest I've got is by using 3 points in this equation - $polygon = p1 + u(p2-p1) + v(p3-p1)$. But by using points ...
2
votes
2answers
100 views

parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
1
vote
1answer
77 views

Find the shaded areas A1, A2, A3

I think i know how to find the angles KAG and KAH that's what i did: (this is the picture from my assignment sheet) then i have to find the shades areas A1,A2 and A3 but i don't know how to ...
1
vote
1answer
224 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
80 views

sketch the line segment whose parametric equations

Sketch the line segment whose parametric equations are $x=2+t, v=t^2-1, t∈[0,3]$ That's what i did $t=x-2$ $v=(x-2)^2-1$ $v=x^2-4x+3 $ $v=(x-3)(x-1)$ $x=3,1$ and that's my sketch I am not ...
0
votes
1answer
127 views

Find a vector equation and parametric equations or the line in R^3 that passes through the point (1,2,-3) and is parallel to the vector u=(4,-5,1).

Find a vector equation and parametric equations or the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
0
votes
1answer
155 views

Find a vector equation and parametric equations of the line in $\mathbb{R}^2$ passing through the origin and is parallel to the vector $\vec{u}=(2,3)$

anyone can help me? :< Are there any equations that I could use in this question? I am so confused. I only know how to do the question if it changes "parallel" to "perpendicular" because I only ...
0
votes
2answers
162 views

Parametric equations and specifications of a triskelion (triple spiral)

I haven't been able to find the parametric equations and specifications to form a triskelion, a triple spiral (this is made of three interlocked couples of spirals). Using the parametric equation of ...
0
votes
2answers
96 views

Cartesian equation of $ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $

I have this parametric equation: $$ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $$ and I have to obtain the Cartesian equation. Any ...
2
votes
1answer
56 views

Area of a band in $\mathbb{R}^2$

If I have a continuous, and smooth curve $\mathcal{C}$, length $\ell$, in $\mathbb{R}^2$ and at each point on the curve I were to draw a line segment, length $d$, normal to the curve centered at the ...
-1
votes
2answers
126 views

Identifiying the next point on the surface of a cube ( or 3D object )

I have a cube of unit length. Each face of the cube is divided into 10 x 10 equal segments. Consider an object of size equal to that of a segment moving through the surface of the cube ( or any 3D ...
0
votes
0answers
48 views

symmetric ranges of curve division on an oloid

taken from wikipedia , I drew an Oloid by using the functions with ; ; and with ; ; If I use any equal spaced range of t(x) on ; I will end up with an unequal range t(y), which ...
2
votes
6answers
256 views

Parabola in parametric form

Show that the following system of parametric equations describes a line or a parabola: $$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
0
votes
0answers
33 views

Strange attractors - groups other than four?

I'm using the following code to generate strange attractors: ...
0
votes
2answers
96 views

What is the $uv$ pair, or $uv$-plane, exactly?

Maybe the answer to this question is easier than computing $1+1$, but I often find this $uv$ pair on pretty much all the parametric equations that have something to do with 3D geometry and all the ...
0
votes
2answers
61 views

Ray-Lens Intersection

So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector. I also have a spherical lens with aperture ...
0
votes
1answer
109 views

Ray Disk intersection

So if I have a ray parameterized as $O + tD$ where $O$ is the origin, $D$ is the direction and $t$ is the parameter variable and a flat circular disk with a center point $P$ in 3D space and a radius ...
0
votes
1answer
36 views

Find position on surface of a lens

If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following ...
0
votes
1answer
29 views

How to check the visibility of these three points?

For question d part i, I have calculated the distances from $Q$ to $P_1$ and $P_2$ respectively and found $P_1$ to be closer with a distance of root $6$, with $P_2$ having a distance of root $24$. ...
0
votes
0answers
72 views

Torus equation in terms of tangent

So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
0
votes
1answer
64 views

3D Surface parametrization basics

I'm studying 3D rendering: I have a surface and the points on the surface are given by some function f such that $p = f (u, v)$ Since I'm a newbie this is unclear to me: how can a function of u and v ...
0
votes
1answer
102 views

parabola in homogeneous coordinates

So if I have the parabola Y = X^2, how do I go about representing this homogeneously? I know I can parameterize it as F(t) = (t, t^2), but then what? The reason I ask is because I have a 3*3 matrix ...
2
votes
0answers
642 views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
1
vote
1answer
2k views

Equation for making a circle in 3D space

I have a 3D space with axis $(x, y ,z)$ and I can make a circle in the $xy$-plane. To make a circle in the xy-plane I currently use spherical coordinates $(r, \theta, \phi)$ where $r = 1$, $\theta = ...
1
vote
2answers
233 views

A circle on the plane [duplicate]

Possible Duplicate: Parametric Equation of a Circle in 3D Space? I know that, for example, if a circle is on a plane with counter-clockwise orientation, and with center $(a,b)$ and radius ...
2
votes
2answers
315 views

Distance between point and a spiral

I'm trying to work out an algorithm where, given the equation for a spiral in polar coordinates, $r(\theta)$, and a point rectilinear coordinates, $P(x,y)$, I can work out the minimum distance between ...
0
votes
1answer
236 views

Is this valid parametric equation to create control points for a helix in 3D space?

Is this a valid way to compute new points that are on a helix and if not what is it wrong? The Cartesian coordinates of each new helix control point could be described by the following ...
0
votes
0answers
863 views

Converting standard equation for a paraboloid to a parametric one

I have the equation for a hyperbolic paraboloid in $x$, $y$, and $z$: $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$ I also have the parametric equations for the same parabaloid: $$x = a u ...
2
votes
2answers
2k 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 ...
2
votes
0answers
909 views

Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
2
votes
1answer
768 views

Equation for control point distance for fixed-length cubic Bézier path (with specific constraints)

A particular Stack Overflow question asks how to construct a specific cubic Bézier path of constant length. I have experimentally determined the ideal distances of the control points from the nearest ...
2
votes
1answer
176 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
2answers
413 views

Finding a quadratic Bézier curve of length $l$ between two points

I have two points $P_1$ and $P_2$ in the plane. For each of the points, I have two vectors $v_1$ and $v_2$. I want to find a quadratic Bézier curve from $P_1$ to $P_2$ of length $l$ leaving $P_1$ in ...