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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9answers
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Is this Batman equation for real?
HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
33
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4answers
1k views
A circle rolls along a parabola
I'm thinking about a circle rolling along a parabola. Would this be a parametric representation?
$(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$
A gives us the radius of the circle, B changes the frequency ...
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11answers
5k views
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 $x$ ...
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vote
2answers
358 views
Parallel functions.
In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line.
I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...
7
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5answers
1k views
A Math function that draws water droplet shape?
I just need a quick reference. What is the function for this kind of shape?
Thanks.
7
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1answer
990 views
Parametrizing implicit algebraic curves
Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have Lockwood and Zwikker to thank). I learned early on that for the purposes of generating plots on a ...
4
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1answer
693 views
Why does the focus of a rolling parabola trace a catenary?
I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
2
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2answers
81 views
What is the limit distance to the base function if offset curve is a function too?
I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
2
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4answers
409 views
Sketch a curve given parametrically by $x=2t-4t^3$ and $t^2-3t^4$
I am unable to see how to eliminate $t$. Wolfram Alpha fails at it too.
$$x=2t-4t^3$$ $$y=t^2-3t^4$$
I can guess that the curve is a polynomial equation so in principle I can write this as
$$w_1 ...
4
votes
1answer
105 views
Assigning alternate crossings to closed curves
This is a minor curiosity that I've been wondering about. Suppose that we draw a closed curve in the plane and that this curve intersects itself several times, but never twice in one spot. We can knot ...
2
votes
1answer
173 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 ...
2
votes
2answers
326 views
Reconstructing space curves from its curvature and torsion
I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
2
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1answer
103 views
Does using an ellipse as a template still produce an ellipse?
Suppose I have a (physical) template, consisting of a piece of stiff sheet plastic with a hole cut in the middle. Suppose the hole is in the shape of an ellipse, say, 8 x 12 inches. Suppose I then ...
2
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2answers
372 views
Direction of the second derivative of an arclength parametrized curve
I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
11
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5answers
4k views
Drawing heart in mathematica
It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer ...
7
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1answer
429 views
Comprehensive compilation of conic section formulae
My frustration started after hours of searching failed to turn up a formula for the vertex of a parabola in the general form
$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
As is already well known, the discriminant ...
3
votes
3answers
1k views
Writing a Polar Equation for the Graph of an Implicit Cartesian Equation
If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$.
Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$.
But I don't know what to do next, since ...
6
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2answers
426 views
Deriving an implicit Cartesian equation from a polar equation with fractional multiples of the angle
Usually, applying the conversion formulae
$r^2=x^2+y^2$
$\cos\;\theta=\frac{x}{r}$
$\sin\;\theta=\frac{y}{r}$
to transform an equation in polar coordinates to an implicit Cartesian equation is ...
5
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1answer
490 views
Formula for curve parallel to a parabola
I have a simple parabola in the form $y = a + bx^2$. I would like to find the formula for a curve which is parallel to this curve by distance $c$. By parallel I mean that there is an equal distance ...
3
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1answer
311 views
The area of the superellipse
I'm watching this video, where D. Knuth explains the connection of $\pi$ and factorials, and other matters (it is very interesting). Almost at the end of the talk he says the area of the superellipse
...
3
votes
1answer
254 views
Why do definitions of distinct conic sections produce a single equation?
I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
3
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1answer
565 views
Can you write a non-piecewise equation that describes an arbitrary shape?
This batman equation thing got me thinking: for an arbitrary curve drawn on the Cartesian plane, can you write a corresponding equation which is not piecewise? What about closed shapes, a la the ...
2
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0answers
60 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 ...
2
votes
1answer
280 views
Why does an equiangular spiral become logarithmic (intuitively)?
One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all ...
8
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2answers
494 views
What type of curve is $\left|\dfrac{x}{a}\right|^{z} + \left|\dfrac{y}{b}\right|^{z} = 1$?
In order to fit experimental data, I want to use a Cartesian equation which looks like:
$\left|\dfrac{x}{a}\right|^{z} + \left|\dfrac{y}{b}\right|^{z} = 1$
$a$, $b$, $c$, and $z$ can take any real ...
4
votes
1answer
456 views
Arc length formula for the lemniscate
This question can be homework for elementary calculus.
The lemniscate of Bernoulli $C$ is a plane curve defined as follows.
Let $a > 0$ be a real number.
Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be ...
2
votes
1answer
335 views
2
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2answers
293 views
A hyperbola as a constant difference of distances
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 $2a$, the distance between the two ...
1
vote
3answers
266 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 ...
1
vote
1answer
177 views
How to place objects equidistantly on an Archimedean spiral?
To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects.
This is what I have so far: The length of a spiral is ...
0
votes
0answers
34 views
Curve is approximatable by function
I want to show that for $\gamma: [a,b] \subseteq \mathbb{R} \rightarrow V$ continously differentiable where V is a bounded subset of $\mathbb{R}^2$. There is always a sequence of functions ...
0
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1answer
56 views
Representing a curve as a plane curve in different ways
Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$.
I know that $X$ has a plane model. More ...
0
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1answer
133 views
A question about geometry of plane curve books
I can't study algebraic geometry yet, so before i learn that, yet i need a comprehensive treatment to the geometric theory of plane curves. i notice that the books refer to this subject either out of ...
0
votes
3answers
258 views
points toward the center of the osculating circle (second derivate in a arc length parameter curve)
Is it true that if I have a planar curve parametrized by its arc length, then the second derivative points towards to the center of the osculating circle?
I can´t see it, but the book says that it´s ...
0
votes
4answers
176 views
how can I graph a bicorn given only its equation?
what are the parts or the variables present in the bicorn equation?
