plane-curves

If C⊆P2C \subseteq \mathbb{P}^2 is a plane curve, then genus(C)=12(d−1)(d−2)genus(C)=\frac{1}{2}(d-1)(d-2). Compare with example in the notes

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In my Algebraic Geometry notes (see http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) there is the following exercise: If C⊆P2 is a plane curve of degree d, then its arithmetic genus g(C) is equal to 12(d−1)(d−2). Compare this to example 0.1.3 (of the notes of course). In fact, the first part of the exercise is easy to solve using the Hilbert … Read more

Characterizations of cycloid

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There are several motions that create a cycloid. I have some examples here. Are there any others? Trace of a fixed point on a rolling circle Evolute of another cycloid (the locus of all its centers of curvature) Involute of another cycloid (trace of a pendulum constrained to another cycloid) Envelope of a family of … Read more

Circle of radius of Intersection of Plane and Sphere

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The plane x+2y−z=4 cuts the sphere x2+y2+z2−x+z−2=0 in a circle of radius? I tried putting value of y from plane in sphere but then I get a zx term. How to proceed? Answer First a normal vector to the plane is (1,2,−1). The sphere equation could be rewritten as (x−12)2+y2+(z+12)2=52, hence the center is (12,0,−12). … Read more

Curve in osculating plane if acceleration is constant

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The book Second Year Calculus: From Celestial Mechanics to Special Relativity makes this comment that if acceleration vector is constant, then the curve lies in the osculating plane spanned by unit normal and unit tangent. The book generally proves its claims but this is just given as comment. So I think there is something very … Read more

What will the equation of a sine wave having the axis as a parabola y=ax2y= ax^2 be?

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approximate representation of my curve(in red) Answer \newcommand{\vect}[1]{{\bf #1}} Define a point on the parabola through a parametric equation \vect{x}(t) = {t \choose at^2} The tangent vector to the parabola is \vect{v}(t) = \frac{{\rm d}}{{\rm d}t}\vect{x}(t) = {1 \choose 2at} and a perpendicular vector can be constructed from this \vect{u}(t) = {2at \choose -1} moreover, … Read more

How does one get from smooth, projective, algebraic curve of genus one to the usual equation for elliptic curves?

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I am used to understanding elliptic curves as a non-singular curve over some field given by the equation $$ y^2 = x^3 +ax + b. $$ However, I have also seen that elliptic curves can be characterized as a smooth, projective, algebraic curves of genus one. Clearly the former definition satisfies this characterization, but I … Read more

Let J⊂R2J \subset \Bbb{R}^2 be homeomorphic to a circle. By the JCT, J separates R2\Bbb{R}^2 into two components. Is the bounded one simply connected?

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Question in the title. If you separate the plane with a Jordan curve, is the bounded component of its complement simply connected? Intuitively, you would think that the curve might extend to a homeomorphism of the 2-disc, with the image of the interior contained within the bounded component. Answer This is the Schoenflies problem for … Read more