Module III·Article III·~1 min read
Algebraic Curves of Higher Orders
Conic Sections
Turn this article into a podcast
Pick voices, format, length — AI generates the audio
Algebraic Curves
Curves of Higher Degrees
Cubic curves: $Ax^3 + Bx^2y + Cxy^2 + Dy^3 + \ldots = 0$. Examples: Descartes' cubic, Bernoulli's lemniscate ($r^2 = a^2 \cos 2\theta$), strophoid.
Lemniscate: $(x^2 + y^2)^2 = a^2(x^2 - y^2)$. Area $S = a^2$. Connection with elliptic integrals: the length of the lemniscate is expressed through an elliptic integral of the first kind — historically, this motivated Gauss's theory.
Transcendental Curves
Cycloid: $x = R(t - \sin t)$, $y = R(1 - \cos t)$ — the trajectory of a point on a rolling circle.
Length of one arch: $8R$. Area under the arch: $3\pi R^2$ (three circles).
The cycloid is the solution to the brachistochrone problem (the path of shortest descent time under gravity): Johann Bernoulli in 1696. The problem led to the development of the calculus of variations.
Curves in Polar Coordinates
Rose: $r = a \cos(n\varphi)$ — $n$ petals for odd $n$, $2n$ for even.
Spirals: Archimedean ($r = a\varphi$), logarithmic ($r = ae^{b\varphi}$), Fermat's ($r^2 = a^2\varphi$).
The logarithmic spiral is self-similar: when rotated by any angle it maps onto itself (only the scale changes). Found in nature: nautilus shells, the arrangement of seeds in a sunflower, galaxies.
§ Act · what next