# Tagged Questions

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

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### What is my line integral answer incorrect?

EDIT: Is my computation not correct, possibly because the parametrization that I used requires x,y to be on the xy-plane? If so, can I adjust from here, and not start over? I.e., is there some ...
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### Help in showing that an evolute is the envelope of the normals to a curve

Let $\alpha:I\to R^2$ be a regular parametrised plane curve (arbitrary parameter), and define $n=n(s)$ and $k=k(t)$ to be the normal and curvature respectively. Assume $k(t)\neq0$, $t\in I$. In this ...
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### 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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### Parametrization of Hyperbola

I "know" that a parametrization of an Hyperbola ($x^2-y^2=1$) is given by: $$\gamma(t)=(\sec(t),\tan(t)),t\in\mathbb{R}$$ I know that $x=\sec(t)$ and $y=\tan(t)$ is a solution of the equation. How ...
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### When working with trochoids what does θ stand for?

These are the formulas with which you can draw trochoids. $x = aθ - b sin(θ)$ $y = a - b cos(θ)$ I'm trying to make trochoids but I got hung up on this symbol $θ$, what is it and how do I use it, I ...
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### How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
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### Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the ...
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### Fitting an ellipse to a point with the first and second derivatives specified

I am trying to fit an ellipse to the end of a curve, $y(x)$, such that first and second derivatives of the curve, $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, are preserved at the contact point, $(x,y)$, ...
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### move curve normal to itself

I have a plane curve given by $y = f(x)$. At every point on this curve, I construct a normal direction to this curve and move the point a fixed distance $s$ along the normal to a new point. What is ...
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### bending an arc to accommodate a constraint

I'm working with piecewise polynomial spirals: curves of the form $z(t) = z_0 + \int_0^t e^{i f(s)} ds$ where $f$ is a quartic polynomial determined by the tangent angles and curvatures at given ...
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### Existence of a Rectifiable Piecewise Smooth Path

Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise ...
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### Metric on the space of plane curves

I am looking for a metric $d$ for smooth 2D curves. Hence $d(x,y)$ is the distance between the curves x and y. For the moment, we may assume that $x$ and $y$ are just directed line segments. Do you ...
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### Piecewise Smooth Curve

Is the curve defined by: γ(t) = (t,t) for 0≤t≤1 and (2-t,2-t) for 1≤t≤2 piecewise smooth? My logic says yes because one can break it into a finite number of smooth curves (two in this case), but ...
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### Green's theorem and 'simple regions'

I'm looking through at my notes and couldn't understand why, in the notes below, there is a need to compute the curve C2 and C4. It's hard to see why isn't computing C1 and C3 a sufficient ...
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### Equation perpendicular to 2 non-parallel planes

Good day sirs! Can you help me with this questions? Find the general equation of the plane: (1) Through $(3,0,-1)$ and perpendicular to each of the planes $x-2y+z=0$ and $x+2y-3z-4=0$ (2) ...
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### 2D trajectory in minimum amount of time given min/max acceleration per axis

I am having a little problem with determining a trajectory. I have a 2D curve, say $\{x(\lambda), y(\lambda)\}$ where $\lambda$ is not time; it is only a parameter that describes the evolution of the ...
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### Finding the equation for the tangent plane to earth given latitude and longtiude

I'm creating a program where I need to calculate the equation of the plane tangent to the earth at a given latitude and longitude. I used Projecting an Arbitrary Latitude and Longitude onto a Tangent ...
A parametrization of a 1-dimensional curve is called regular if its velocity is always positive. For example, the following parametrization: $$x(t)=t^3, y(t)=t^6$$ is not regular because its ...