Analytic geometry

2 artículos

Cartesian coordinates Equations of lines Parabolas Ellipses Hyperbolas Polar coordinates Intercept theorem Parametric curve Parametric surface Polar coordinate system Analytic curve Analytic surface Bianchi surface Canal surface Canonical curve Cantor curve Cayley surface Ces ro curve Covering surface Curve of constant slope Pursuit curve Cycloidal curve Cylindrical surface Demoulin surface Discriminant curve Evolvent of a plane curve Focus of a curve Fr chet surface Fundamental forms of a surface General type algebraic surface Genus of a curve Genus of a surface Hessian Integral surface Iso optic curve Isothermal surface Joachimsthal surface Klein surface Liouville surface Menger curve Moulding surface Persian curve Peterson surface Plane real algebraic curve Pontryagin surface Pr fer surface Quadratic surface forms Regression surface Rotation surface Saddle surface Schwarz surface Surface of screw motion Surface potential Swerve of a curve Tangent curve of the Transport surface Two sided surface Viviani curve Voss surface Weingarten surface

Analysis / Calculus GradoAnálisis9 min

Root Pi from a Bell Curve, by Leaving the Number Line

There is no elementary antiderivative to evaluate, and the checkable slice of that is one line: if p is a polynomial then p' - 2xp has degree deg p + 1, which can never equal the degree of 1. Squaring the integral turns it into a rotationally symmetric integral over the plane, where the polar area element supplies the factor r that makes the radial integral elementary, so I squared equals 2 pi times one half. The same idea survives without polar coordinates via the substitution y = xt, and it fails for e to the minus x to the fourth because x^4 + y^4 is not a function of the radius.

Geometry BachilleratoExplicación9 min

Outrunning a Dog Four Times Faster, on the Margin Between Pi and Three

Running straight out from the centre loses, because a radius costs you 1 while half the fence costs the dog pi over 4. Inside a quarter of the radius your angular speed beats his, so you can orbit until he is diametrically opposite and then sprint three quarters of a radius against his pi over 4, and the whole escape reduces to 3 being less than pi with a margin of 0.0354 R. That two-phase plan works only up to a speed ratio of pi + 1, while the best known strategy for the problem reaches 4.60334.