1
vote
2answers
74 views

Retrieve the initial cubic Bézier curve subdivided in two Bézier curves

I have a cubic Bezier curve subdivided to two cubic Bezier: Assuming that "t_cut" is the t value where this initial Bezier is cut: example of function subdivision(BezierCurve initialCurve, ...
4
votes
0answers
77 views

Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
3
votes
1answer
48 views

Paths followed by Morley triangle vertices as apex moves parallel to base

Let the vertices of a triangle $T$ be $(A,B,C)$, and $(a,b,c)$ the vertices of its Morley triangle $M$. Designate vertex $C$ as the apex of $T$. Now move apex $C$ parallel to $AB$, all the while ...
0
votes
1answer
21 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
2
votes
0answers
76 views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
5
votes
1answer
124 views

How can a simple closed curve not look locally like the rotated graph of a continuous function?

A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ...
0
votes
1answer
48 views

The area of a region around a curve

If we are given a simple closed curve with length $L$ in the plane, and we have a fixed number $r$ such that for each point $x$ on the curve there is a related disc $D(x,r)$ with radius $r$, how can ...
1
vote
2answers
41 views

Find the distance of the point $( 1,2,3 )$ to ,,,,,,,,,,,,,,

Find the distance of the point $( 1,2,3 )$ of the plane $3x-2y+5z+17=0$ .
1
vote
0answers
27 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
2
votes
1answer
91 views

Line tangent to curve in complex plane

For what pairs $a,b\in\mathbb{\mathbb{C}}$ is the line $L(x,y)=ax+by+1=0$ tangent to the curve $C(x,y)=x^4+y^4+1=0$? By definition of "tangent", if I have a point $(c,d)\in C$, and a line ...
5
votes
1answer
95 views

Relations between curvature and area of simple closed plane curves.

Let $\gamma$ be a simple closed plane curve. We know that a curve with constant curvature $\kappa$ will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ...
1
vote
1answer
32 views

How to determine a point is outside or inside

How could I determine a point is outside or inside of a domain with variable raduis. like this: $$x(t)=(0.3+0.2(\sin3t))\cos t$$ $$y(t)=(0.3+0.2(\sin3t))\sin t$$ where 0$\leq t< 2\pi$. I tried ...
5
votes
0answers
70 views

Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. ...
1
vote
1answer
46 views

Minimum distance from curve

I was thinking about the following problem: Let $\gamma \subset \mathbb R ^2$ be a curve that admits a $C ^{\infty}$ regular parametrization. Is it always possible to choose an open set $E$ ...
1
vote
0answers
89 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
0
votes
1answer
66 views

$d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$

If $f: S_1\to S_2$ is a smooth map betweentwo regular surface $S_1$ and $S_2$, then $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$ I cannot prove this statement. I think that ...
0
votes
0answers
65 views

Inverse of a pythagorean hodograph curve.

I'm trying to solve this question about pythagorean hodograph curves: If $\alpha$ is a Pythagorean hodograph curve, and $\alpha(t)\neq(0,0)\;\forall t$, then the curve $\frac{1}{\alpha(t)}$ is also a ...
1
vote
1answer
70 views

Curve described by a point inside an ellipse

It's well known that a point inside a circle rotating on a line describes a trochoid of parametric equation: $$x=c_0\phi-c_1\sin(\phi)$$ $$y=c_2-c_3\cos(\phi)$$ in which the constant ...
4
votes
0answers
105 views

In how few points can a continuous curve meet all lines?

Inspired by this (currently closed) question, I'm wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for ...
3
votes
1answer
338 views

Minimize the sum of distances between two point and a circle

Let's $A$,$B$ and $O$ be random point in a plane, such that they are not colinear. Let's $c$ be a circle centered on $O$, such that points $A$ and $B$ are outside of it. Find a point $X$ that lies on ...
2
votes
1answer
165 views

Support function of a convex domain

Let $Q$ be a compact, convex domain in the plane, with smooth boundary $\partial Q$. We further assume that the origin is contained in $Q$. For a concrete example, let's take an ellipse ...
1
vote
1answer
122 views

Parallel to line on $f(x)=1+\sin(x)/x$

I want to draw a curve on the top of the function $f(x)=1+\dfrac{\sin(x)}{x}$, but the curve should be equidistant (perpendicular distance from any point of the function $f(x)=1+\dfrac{\sin(x)}{x}$) ...
2
votes
0answers
49 views

Jordan curves, its interiors and the existence of a continuous function.

Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ is a Jordan curve, $O(t)$ is its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, is there exists a continuous ...
3
votes
3answers
677 views

How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?

Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$ My book says the answer is $(0,0),(a,0),(a,\pi)$. However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
8
votes
1answer
153 views

Is it possible for a Jordan curve in the plane to enclose a set with area zero?

I read about the Isoperimetric Inequality the other day. It says that for any Jordan curve, $$ \frac{4 \pi A}{L^{2}} \leq 1, $$ where $ L $ is the length of the curve and $ A $ is the area of the ...
2
votes
1answer
331 views

What does the definition of curvature mean?

First question, am I right in saying that curvature measure how quickly the direction of a cruve changes? Also, we have been given the "definition": $$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$ where ...
3
votes
1answer
499 views

Using equations to draw out complex objects

How do people come up with equations of curves to draw out complex objects? Some popular examples would include: batman curve & PSY curve. This stackexchange link explains the rationale for the ...
32
votes
5answers
1k views

Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
1
vote
0answers
194 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
1
vote
1answer
58 views

Determine the path that created by a sector of circle

ABC circle sector turns on ground (x axis) as shown in the figure. A is the center of the circle. $\angle{OAB}=\angle{OAC}=\alpha $ $|AB|=r$ $\cfrac{|AP|}{|PC|}=k$ The corners meet on point $H$ ...
4
votes
2answers
256 views

Locus of points generates several very different curves. Closed form?

Consider, for the sake of simplicity, a circle $C$ centered at he origin with radius $a$. Let $F=(h,k)$ be a point not necessarily inside the circle. Let $M=(a\cos\theta,a\sin\theta)$ be a point in ...
3
votes
1answer
789 views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
1
vote
3answers
246 views

Looking for the curve traced by a moving bicycle when its steering wheel is fully rotated

I am looking for a curve traced by a moving bicycle when its steering wheel is fully rotated either clockwise or anti-clockwise. How to model it mathematically? Is the curve a circle? My attempt is ...
8
votes
2answers
591 views

Fireworks under inverse-cube gravity

What is the path of a projectile under an inverse-cube gravity law? Imagine that the law of gravity was changed overnight from $F(r) = G m_1 m_2 / r^2$ to $F(r) = G' m_1 m_2 / r^3$. To be ...
8
votes
2answers
229 views

Where do people learn about things like caustics, evolutes, inverse curves, etc.?

When I look up a curve on Wikipedia, I'll often see a lot of properties along the lines of "you can generate curve X by rolling a circle along curve Y and tracing the trajectory of a single point," or ...
3
votes
1answer
288 views

Name that curve!

What curve will a kayak describe if the paddler aims her bow at an object on a distant shore ahead and keeps the bow pointing to that object as she paddles toward it with constant velocity, in the ...
2
votes
0answers
85 views

Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
5
votes
4answers
3k views

How to find the parametric equation of a cycloid?

"A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia In many calculus books I have, the cycloid, in parametric form, is ...
3
votes
2answers
136 views

How to mathematically color the regions bounded by a parametric curve?

Usually, if an implicit equation F(x,y) = 0 defines a curve (or curves) on the x-y plane, then we can use the inequalities F(x,y) < 0 or F(x,y) > 0 to color the regions bounded by the curve (or ...
0
votes
2answers
900 views

How to draw a family of curves and its envelope?

Given a family of curves $$F=(x-t)^2+y^2-\frac{1}{2}t^2,$$ I am trying to compute the envelope of this family. The envelope is described by the equations $$F=0, \\ \dfrac{\partial F}{\partial t} ...
0
votes
2answers
71 views

Is there plane curves with limit number of operations in which is non-constructible and how do we prove it

Is there plane curves with limit number of operations in which is non-constructible and how do we prove it is non-constructible, i call it non-constructible if we have to plot infinity number of point ...
4
votes
1answer
703 views

Connect two curves with Euler spiral segment

Image of situation: http://upload.wikimedia.org/wikipedia/commons/5/54/Easement_curve.svg Let's say we have a straight line (blue) and a circular arc (green). My goal is to connect these two curves ...
2
votes
3answers
431 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
8
votes
1answer
737 views

What are curves (generalized ellipses) with more than two focal points called and how do they look like?

An ellipse is usually defined as the locus of points so that sum of the distances to the two foci is constant. But what are curves called which are defined as the locus of points so that the sum of ...
0
votes
1answer
850 views

Making a circle with paper folding, scissors, pencil, and a straightedge

Can we make a circle using paper folding, scissors, straightedge, anda pencil, allowing an infinite number of operations? I think my chemistry teacher have show me once how to make it during the ...
5
votes
1answer
347 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
1
vote
1answer
101 views

Point travels around curve

I wonder what does this mean: Point travels around curve. I try to figure out some math explanation in the book and I can't move forward because I can't understand these words. I can understand when ...
0
votes
1answer
623 views

definition of sinusoidal curve

I have question related with these two definition: In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates $$r^n = a^n \cos(n \theta)$$ where $a$ is ...
1
vote
1answer
363 views

area bounded by spirogram

A circle of radius r rolls without slipping inside an n-gon of side length l. A curve C is traced out by a pencil through a hole a distance d from the centre. Initially the circle is in a corner with ...
2
votes
2answers
366 views

Ways to define a curve

I'm trying to give shapes in my physics engine roundness/ curvature. I am aware of various methods for mathematically defining curvature such as bezier-curves, ellipses, etc; but I'm not sure which ...