# 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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### Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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### The plane curve of maximum average speed under constant gravitational force

A line gives us the minimum distance from $A$ to $B$. A cycloid gives us the minimum traveling time of a point mass from $A$ to $B$ (under constant gravitational acceleration $g$). What about the ...
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### Measure a pyramid or prism [closed]

Is it more challenging to measure a pyramid or prism? Discuss, making clear what attributes you are measuring, and whether you are determining measurements physically or through calculation. My ...
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### Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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### 3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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### What is the definition of a path along a multivariable function?

I'm taking a class equivalent of Calculus III, and we saw how to prove continuity of a multivariable function. Recently we looked at the following example: \begin{align} f(x,y) = \begin{cases} \frac{...
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### S-shaped Reverse Logistic Curve

Is there any curve that grows very slow at the beginning then growth picks up exponentially before hitting the wall. I need sort of reverse behavior of the logistic curve. Here's raw idea how it ...
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### Equation of a plane from 2 lines

I have two lines with the following equation $$D1 : (x, y, z) = (2,0,0) + k(0,3,0)$$ $$D2 : (x, y, z) = (2,0,2) + k(0,0,1)$$ and I must find out the equation of the plane that they make. I ...
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I'm trying to learn how can I write the equation for a line given all the points that belongs in the line. I'm looking to find the equation for a curve. An example set of points is: $$\{ (24,11), (... 11answers 406k views ### Is this Batman equation for real? HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? 1answer 21 views ### Find volume of cube with the help of eqn of plane The volume of cube whose two faces lie on the plane 6x-3y+2z+1=0 and 6x-3y+2z+4=0? 1answer 36 views ### Level curve of the function f(x,y)=\min\{x^2+y^2,xy\} [closed] How I can find the level curve of the function$$f(x,y)=\min\{x^2+y^2,xy\}$$From where I need to start to solve this problem? Thank you! 1answer 22 views ### Path from a start point at a certain heading to an end point at a certain heading while obeying a minimum turn radius So not sure if my title is clear (also no idea what to tag, because you need 1000 rep to add tags) so I will try my best to explain the problem. I'm working in 2D space and to simplify the problem, I'... 3answers 297 views ### How to find center of a conic section from the equation? If we are given a curve in the form$$ax^2+2bxy+cy^2+2dx+2ey+f=0$$and the following determinant$$\delta=\begin{vmatrix}a&b\\b&c\end{vmatrix}=ac-b^2$$is non-zero, then this is either a curve ... 2answers 44 views ### Injective curve How can i show that the curve \gamma:\mathbb{R}\to\mathbb{R}^2 defined by $$\gamma(t)=\left(\frac{t}{1+t^2},\frac{t}{1+t^4}\right)$$ is injective (without using algebraic ... 1answer 38 views ### Does there exist a curve with non zero area? Does there exist a curve with a bounded infinitely diffrentiable derivative (i.e. has a minimum |curvature|) of hausdorf demension 2? Or even a diffrentiable curve of hausdorf demension 2? 2answers 71 views ### Index of a Jordan curve Winding number theorem: If J\subset \mathbb{C} is a Jordan curve and a point z lies in its interior domain, then the winding number n(J,z)=\pm 1. Now suppose that J is smooth and we have the ... 0answers 36 views ### Blowing up a model for a plane curve Let R = \mathbb{C}[[t]] and let \mathcal{X} \hookrightarrow \mathbb{P}_R^2 be the arithmetic surface defined by the equation$$(X^2 - 2Y^2 + Z^2)(X^2 - Z^2) + tY^3Z = 0.$$The generic fiber ... 1answer 44 views ### How to prove this inequality about the arc-lenght of convex functions? Let be f,F:[0,1] \to \mathbb{R} with f,F \in C^2([0,1]) two convex functions such that f \le F in all points and f(0)=F(0), f(1)=F(1). Considering the plane-curves \gamma(t)=(t,f(t)) and ... 1answer 127 views ### converse to the jordan curve theorem Suppose K\subset \mathbb{R}^2 is compact and locally connected, and does not contain 0. Let A be the component of \mathbb{R}^2-K containing 0, and let B be the unbounded component. Assume ... 1answer 32 views ### find a closed curve with given winding numbers Is there a closed (piecewise) C^1 curve \gamma: [a,b]\rightarrow\mathbb{C}, so that \mathbb{C}\setminus\gamma([a,b]) has four components with winding numbers -1, 0, 2, 3? Thanks! 5answers 344 views ### Parametric equation of x^y=y^x curve What is the easiest/most natural way to parametrize the following curve?$$\{(x,y)\in\Bbb R^{+2}\mid x^y=y^x, x\neq y\}\cup\{(e,e)\}$$The best I could do was taking it apart, and for x>y use ... 0answers 34 views ### Which curve (surface) is this? We're having trouble fitting our data... well, we don't even know which function we should fit onto. Anybody knows if this function is well defined mathematically? 0answers 35 views ### A curve with Lebesgue measure non zero In this Continuously Differentiable Curves in \mathbb{R}^{d} and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact ... 1answer 24 views ### Log power rule problem According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian ... 0answers 35 views ### Roots of the Taylor approximation of the exponential While answering another question, I looked at the roots of the n^{th} degree Taylor approximation of the exponential.$$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$Apparently, these root are ... 2answers 34 views ### Parametric version of a simple equation I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : \... 1answer 60 views ### Find the intersection of plane and sphere If the equation of the sphere is x^2+y^2+z^2=1 and the plane is x+y+z=1, then how can the equation of a circle be determined from the equations of a sphere and a plane? and what is the parametric ... 0answers 25 views ### A space curve with non-vanishing curvature is planar iff its torsion is 0 Intuitively this is simple and to prove the backwards direction: \tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b}  is constant. Then letting \gamma be the parametrisation of the curve ... 0answers 11 views ### Exercise concerning areas inside closed curves Let \alpha (s), s\in[0,l], be a closed, convex, plane curve with \kappa >0. Let r be a positive constant and define \beta (s)=\alpha (s)-rn(s), where n(s) is the normal vector of \... 1answer 49 views ### Intersection of sphere and ellipsoid Ellipsoid:$$ x^2+\frac{y^2}{4}+\frac{z^2}{2}=1 $$Sphere:$$ x^2+(y-1)^2+(z-d)^2=1 $$For what values of d, there is a common tangent plane to both curves? Part of my resolution: Consider f(x,y,... 2answers 17 views ### Intersection of sphere and plane is a point, find c, such that (0,0,c) is center of sphere Sphere:$$ x^2+y^2+(z-c)^2=1 $$Plane:$$ x+2y+3z=0 $$Find the values of c, for which the intersection of the sphere and the plane is a point. Well, I know that the sphere has the center (0,0,c) ... 0answers 75 views ### Parametric Interpolation in the Plane Given i+j points in the plane, when can we find x(t),y(t), polynomials of degree i and j respectively such that the parametric curve (x(t),y(t)) goes through each point? We can do this ... 0answers 68 views ### Graph theory: creating surfaces If we draw a circle in the plane , we separate the plane into 2 regions , inside and outside. On the following surfaces, determine the maximal number of circles that you can draw on the surface ... 1answer 44 views ### Finding family of curve for given asymptotes I need to find possible curves, with asymptotes given as x=0 (x \to -\infty) and y=mx \hspace{0.5cm} m>0. it is easy to find curves for individual lines, y= \exp(-\lambda_1 x) + mx for y=mx ... 2answers 51 views ### Closed subset of \mathbb R^2 disconnecting \mathbb R^2 must contain a curve? Edit: Per Alex S's answer, my first statement was false, so I've moved it to the bottom. Here is the weaker statement that I am trying to prove (this was included in my original question): Let C \... 1answer 18 views ### Finding the radii of an ellipse from the intersection of a plane and a sphere I'm trying to solve the following problem, regarding Stokes Theorem: F = z i + xj + yk ; C the curve of intersection of the plane x + y + z = 0 and the sphere x^2 + y^2 + z^2 = 1 [Hint: ... 0answers 46 views ### Dual curve of an algebraic curve in affine coordinates F is an irreductible algebraic curve and we consider the application that sends a non-singular point (a : b : c) in homogenius coordinates, to \phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ... 1answer 29 views ### How to draw cubic plane curve? In Python, using MatPlotLib, given [vector] parameters a and b and [scalar] parameter c, I want to draw a general cubic plane curve in 2-dimensional space (regular plane with x and y axes): ... 2answers 111 views ### What is the Hilbert curve's equation?! The Hilbert curve has always bugged me because it had no closed equation or function that I could find. What is its equation or function? For example, if I wanted to find the Hilbert's curve point at ... 1answer 70 views ### convex curve as boundary of a convex set In short, my question is to know if the following statement is true, and how to show it : Theorem Let \gamma be a closed simple C^2 convex curve in \mathbb{R}^2. We denote \Gamma its image. ... 1answer 54 views ### Path integral over Koch curve If a curve C is a Koch curve (snowflake fractal) centered at origo, then is the following path integral defined?$$\int_C \frac{1}{z}\space dz.$$If it exists, the value must be 2\pi i for ... 0answers 53 views ### Is this method of finding a “dual curve” correct? I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ... 0answers 12 views ### Curve fitting: How to identify the appropriate function for a beat-like phenomena? I have a time series data which shows some beat like behaviour. The envelope does not look exponentially decreasing, as it is impossible from a physics point of view. The envelope is likely to be a ... 0answers 74 views ### Proof of the formula that computes the genus of smooth projective plane curve I was searching for a proof of the formula that computes the genus of a smooth projective plane curve of degree d:$$g = \frac{(d-1)(d-2)}{2} which do not make use neither of triangulation or ...
I'm reading Knuth's Concrete Mathematics and we are dealing with recurrence relations. He proves that the number of regions $L_n$ formed by $n$ lines on a plane is $L_n=\frac{n(n+1)}{2}$. I don't ...