hcju/proofwikips huggingface dataset:
Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a Brouwerian lattice. Let $a$ be an element of $S$. Let $g, d: S \to S$ be mappings such that :$\forall s \in S: g\left({s}\right) = a \to s$ and :$\forall s \in S: d\left({s}\right) = a \wedge s$ Then $\left({g, d}\right)$ is Galois connection.
Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice
Let $\struct {S, \vee, \wedge, \preceq}$ be s Brouwerian lattice. Let $a, x, y \in S$. Then :$a \wedge x \preceq y$  $x \preceq a \to y$
Weierstrass Approximation Theorem
Let $f$ be a real function which is continuous on the closed interval $\Bbb I$. Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy.
Construction of Right Angle by Stretched Rope
Let a rope be knotted at regular intervals so that there are $12$ equal segments separated by knots. Tie the rope in a loop consisting of those $12$ segments. Fix one of the knots to the ground at the point you want the right angle to be placed. Stretch the rope tightly in the direction of one of the legs of the right angle you want to create and fix the fourth knot to the ground. Stretch the rope tightly in the direction of the other leg of the right angle you want to create and fix the third knot in the ground, at the same time making sure the remaining segment of $5$ knots is also tight. The point where the $3$-knot section and the $4$-knot section meet, the rope will be bent at a right angle. :400px
Up-Complete Product
Let $\struct {S, \preceq_1}$, $\struct {T, \preceq_2}$ be non-empty ordered sets. Let $\struct {S \times T, \preceq}$ be Cartesian product of $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$.
Cartesian Product of Ordered Sets is Ordered Set
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets. Let $\left({S_1 \times S_2, \preceq}\right)$ be the Cartesian product of $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$. Then $\left({S_1 \times S_2, \preceq}\right)$ is also an ordered set.
Motion of Rocket in Outer Space
Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\mathbf b$. Then the equation of motion of $B$ is given by: :$m \dfrac {\mathrm d \mathbf v} {\mathrm d t} = - \mathbf b \dfrac {\mathrm d m} {\mathrm d t}$
Velocity of Rocket in Outer Space
Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\mathbf b$. Then the velocity of $B$ at time $t$ is given by: :$\map {\mathbf v} t = \map {\mathbf v} 0 + \mathbf b \ln \dfrac {\map m 0} {\map m t}$ where $\map {\mathbf v} 0$ and $\map m 0$ are the velocity and mass of $B$ at time $t = 0$.
Acceleration of Rocket in Outer Space
Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\mathbf b$. Then the acceleration of $B$ at time $t$ is given by: :$\mathbf a = \dfrac 1 m \left({- \mathbf b \dfrac {\mathrm d m} {\mathrm d t} }\right)$
Up-Complete Product/Lemma 2
Let $X$ be a directed subset of $S \times T$. Then :$\map {\pr_1^\to} X$ and $\map {\pr_2^\to} X$ are directed where  :$\pr_1$ denotes the first projection on $S \times T$ :$\pr_2$ denotes the second projection on $S \times T$ :$\map {\pr_1^\to} X$ denotes the image of $X$ under $\pr_1$
Sum of Cosines of Multiples of Angle
{{begin-eqn}} {{eqn | l = \frac 1 2 + \sum_{k \mathop = 1}^n \cos \left({k x}\right)       | r = \frac 1 2 + \cos x + \cos 2 x + \cos 3 x + \cdots + \cos n x       | c =  }} {{eqn| r = \frac {\sin \left({\left({2 n + 1}\right) x / 2}\right)} {2 \sin \left({x / 2}\right)}       | c =  }} {{end-eqn}} where $x$ is not an integer multiple of $2 \pi$.
Eratosthenes' Measurement of Earth
The circumference of Earth was measured by {{AuthorRef|Eratosthenes of Cyrene}} to be of the order of $45 \, 000 \, \mathrm{km}$.
Sum of Reciprocals of Twin Primes
The sum of the reciprocals of all the twin primes: :$\dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 {11} + \dfrac 1 {13} + \dfrac 1 {17} + \dfrac 1 {19} + \dfrac 1 {29} + \dfrac 1 {31} + \cdots$ is either finite or convergent.
Prime Number Theorem in Eulerian Logarithmic Integral Form
The Prime Number Theorem is equivalent to: :$\displaystyle \lim_{x \mathop \to \infty} \frac {\map \pi x} {\map \Li x} = 1$ where: :$\map \pi x$ is the prime-counting function :$\map \Li x$ is the Eulerian logarithmic integral: ::$\displaystyle \map \Li x := \int_2^x \dfrac {\d t} {\ln t}$
Exponential of Rational Number is Irrational
Let $r$ be a rational number such that $r \ne 0$. Then: :$e^r$ is irrational where $e$ is Euler's number.
Rational Points on Graph of Exponential Function
Consider the graph $f$ of the exponential function in the real Cartesian plane $\R^2$: :$f := \set {\tuple {x, y} \in \R^2: y = e^x}$ The only rational point of $f$ is $\tuple {0, 1}$.
Rational Points on Graph of Logarithm Function
Consider the graph of the logarithm function in the real Cartesian plane $\R^2$: :$f := \set {\tuple {x, y} \in \R^2: y = \ln x}$ The only rational point of $f$ is $\tuple {1, 0}$.
Rational Points on Graph of Sine Function
Consider the graph of the sine function in the real Cartesian plane $\R^2$: :$f := \set {\tuple {x, y} \in \R^2: y = \sin x}$ The only rational point of $f$ is $\tuple {0, 0}$.
Rational Points on Graph of Cosine Function
Consider the graph of the cosine function in the real Cartesian plane $\R^2$: :$f := \left\{ {\left({x, y}\right) \in \R^2: y = \cos x}\right\}$ The only rational point of $f$ is $\left({0, 1}\right)$.
Meet-Continuous iff Ideal Supremum is Meet Preserving
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that: :$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$ where  :$\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$ Then :$\mathscr S$ is meet-continuous  :$f$ preserves meet as a mapping from $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$ into $\mathscr S$
Rational Number is Algebraic
Let $r \in \Q$ be a rational number. Then $r$ is also an algebraic number.
Square Root of 2 is Algebraic of Degree 2
The square root of $2$ is an algebraic number of degree $2$.
Constructible Length with Compass and Straightedge
Let $L$ be a line segment in a Eucldiean space. Let the length of $L$ be $d$. Let $L'$ be a line segment of length $d'$ constructed from $L$ using a compass and straightedge construction. Then: :$d' = q d$ where $q$ is an algebraic number whose degree is at most $2$.
Gelfond-Schneider Constant is Transcendental
The Gelfond-Schneider constant: :$2^{\sqrt 2}$ is transcendental.
Sum of Bernoulli Numbers by Binomial Coefficients Vanishes
:$\forall n \in \Z_{>1}: \displaystyle \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$ where $B_k$ denotes the $k$th Bernoulli number.
Bernoulli Numbers are Rational
The Bernoulli numbers are rational.
Power Series Expansion for Tangent Function/Proof of Convergence
The radius of convergence of the Power Series Expansion for Tangent Function: :$\displaystyle \tan x = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}$ where $B_{2 n}$ denotes the Bernoulli numbers, is given as: :$\size x < \dfrac \pi 2$
Power Series Expansion for Tangent Function/Sequence
The Power Series Expansion for Tangent Function begins: :$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$
Partial Fractions Expansion of Cotangent
Let $x \in \R - \Z$ , that is such that $x$ is a real number that is not an integer. Then: :$\pi \cot \pi x = \dfrac 1 2 + 2 x \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {x^2 - n^2}$
Tangent to Cycloid is Vertical at Cusps
The tangent to the cycloid whose locus is given by: : $x = a \paren {\theta - \sin \theta}$ : $y = a \paren {1 - \cos \theta}$ is vertical at the cusps.
Tangent to Cycloid passes through Top of Generating Circle
Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ Then the tangent to $C$ at a point $P$ on $C$ passes through the top of the generating circle of $C$.
Tangent to Cycloid
Let $C$ be a cycloid generated by the equations: : $x = a \left({\theta - \sin \theta}\right)$ : $y = a \left({1 - \cos \theta}\right)$ Then the tangent to $C$ at a point $\left({x, y}\right)$ on $C$ is given by the equation: :$y - a \left({1 - \cos \theta}\right) = \dfrac {\sin \theta} {1 - \cos \theta} \left({x - a \theta + a \sin \theta}\right)$
Equation of Hypocycloid
Let a circle $C_1$ of radius $b$ roll without slipping around the inside of a circle $C_2$ of (larger) radius $a$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $H$ be the hypocycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$ :$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$
Meet in Set of Ideals
Let $\mathscr S = \left({S, \preceq}\right)$ be a meet semilattice. Let ${\it Ids}\left({\mathscr S}\right)$ be the set of all ideals in $\mathscr S$. Let $I_1, I_2$ be ideals in $\mathscr S$. Then :$I_1 \wedge I_2 = \left\{ {i_1 \wedge i_2: i_1 \in I_1, i_2 \in I_2}\right\}$ where :$I_1 \wedge I_2$ denotes the meet in $\left({ {\it Ids}\left({\mathscr S}\right), \subseteq}\right)$
Intersection of Semilattice Ideals is Ideal
Let $\struct {S, \preceq}$ be a meet semilattice. Let $I_1, I_2$ be ideals in $\struct {S, \preceq}$. Then $I_1 \cap I_2$ is ideal in $\struct {S, \preceq}$
Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii
Let $H$ be a hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$. Let $a = n b$ where $n$ is an integer. Then $H$ has $n$ cusps.
Number of Cusps of Hypocycloid from Rational Ratio of Circle Radii
Consider the hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$. Let $k = \dfrac a b$ be a rational number. Let $k$ be expressed in canonical form: :$k = \dfrac p q$ where $p$ and $q$ are coprime. Then $H$ has $p$ cusps.
Cusps of Hypocycloid from Irrational Ratio of Circle Radii
Consider the hypocycloid $H$ generated by a rotor $C_1$ of radius $b$ rolling within a stator $C_2$ of (larger) radius $a$. Let $k = \dfrac a b$ be an irrational number. Then $H$ has an infinite number of cusps, which are evenly and densely distributed around the circumference of $C_2$.
Equation of Astroid
Let $H$ be the astroid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the inside of a stator $C_2$ of radius $a = 4 b$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. === Parametric Form === {{:Equation of Astroid/Parametric Form}} === Cartesian Form === {{:Equation of Astroid/Cartesian Form}}
Tangent to Astroid between Coordinate Axes has Constant Length
Let $C_1$ be a circle of radius $b$ roll without slipping around the inside of a circle $C_2$ of radius $a = 4 b$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $H$ be the astroid formed by the locus of $P$. The segment of the tangent to $H$ between the $x$-axis and the $y$-axis is constant, immaterial of the point of tangency.
Pendulum Contained by Cycloid moves along Cycloidal Path
Let a pendulum with a flexible rod be suspended from a point $P$. Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs. :400px Then the bob is constrained to move such that its path traces the arc of a cycloid.
Lower Closure of Meet of Lower Closures
Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $D_1, D_2$ be subsets of $S$. Then :$\left\{ {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}\right\}^\preceq = \left\{ {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}\right\}^\preceq$ where :$D_1^\preceq$ denotes the lower closure of $D_1$.
Equation of Cycloid in Cartesian Coordinates
Consider a circle of radius $a$ rolling without slipping along the $x$-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equation: :$a \sin^{-1} \paren {\dfrac {\sqrt {2 a y - y^2} } a} = \sqrt {2 a y - y^2} + x$
Second Derivative of Locus of Cycloid
Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The second derivative of the locus of $P$ is given by: :$y'' = -\dfrac a {y^2}$
Rate of Change of Cartesian Coordinates of Cycloid
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. Then the rate of change of $x$ and $y$ can be expresssed as: {{begin-eqn}} {{eqn | l = \frac {\d x} {\d t}       | r = \mathbf v_0 \paren {1 - \cos \theta}       | c =  }} {{eqn | l = \frac {\d y} {\d t}       | r = \mathbf v_0 \sin \theta       | c =  }} {{end-eqn}} where $\theta$ is the angle turned by $C$ after time $t$.
Maximum Rate of Change of X Coordinate of Cycloid
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The maximum rate of change of $x$ is $2 \mathbf v_0$, which happens when $P$ is at the top of the circle $C$.
Lower Closure is Closure Operator
Let $\left({S, \preceq}\right)$ be an ordered set. Then :lower closure of set is a closure operator.
Area inside Astroid
The area inside an astroid $H$ constructed within a circle of radius $a$ is given by: :$\AA = \dfrac {3 \pi a^2} 8$
Length of Arc of Astroid
The total length of the arcs of an astroid constructed within a stator of radius $a$ is given by: :$\LL = 6 a$
Area of Surface of Revolution from Astroid
Let $H$ be the astroid constructed within a circle of radius $a$. The surface of revolution formed by rotating $H$ around the $x$-axis: :400px evaluates to: :$\mathcal S = \dfrac {12 \pi a^2} 5$
Tusi Couple is Diameter of Stator
A Tusi couple is a degenerate case of the hypocycloid whose form is a straight line that forms a diameter of the stator.
Meet Preserves Directed Suprema
Let $\mathscr S = \struct {S, \preceq}$ be an up-complete meet semilattice such that :$\forall x \in S$, a directed subset $D$ of $S$: $x \preceq \sup D \implies x \preceq \sup \set {x \wedge y: y \in D}$
Equation of Astroid/Parametric Form
The point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin{cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.
Equation of Astroid/Cartesian Form
The point $P = \tuple {x, y}$ is described by the equation: :$x^{2/3} + y^{2/3} = a^{2/3}$
Preceding iff Meet equals Less Operand
Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $x, y \in S$. Then :$x \preceq y$  $x \wedge y = x$
Simple Harmonic Motion of Point on Tusi Couple
Let $C_1$ and $C_2$ be the rotor and stator respectively of a Tusi couple $H$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let the center of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$. Let $P$ be the point on the circumference of $C_1$ whose locus is $H$. Let $C_1$ be initially positioned so that $P$ its point of tangency to $C_2$, located on the $x$-axis. Then $P$ moves back and forward on the $x$-axis with simple harmonic motion with period $\dfrac {2 \pi} \omega$ and maximum speed $a \omega$.
Maximum Rate of Change of Y Coordinate of Astroid
Let $C_1$ and $C_2$ be the rotor and stator respectively of an astroid $H$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let the center $C$ of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$. Let $P$ be the point on the circumference of $C_1$ whose locus is $H$. Let $C_1$ be positioned at time $t = 0$ so that $P$ its point of tangency to $C_2$, located on the $x$-axis. Let $\theta$ be the angle made by $OC$ to the $x$-axis at time $t$. Then the maximum rate of change of the $y$ coordinate of $P$ in the first quadrant occurs when $P$ is at the point where: :$x = a \paren {\dfrac 1 3}^{3/2}$ :$y = a \paren {\dfrac 2 3}^{3/2}$
Equation of Deltoid
The point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin{cases} x & = 2 b \cos \theta + b \cos 2 \theta \\ y & = 2 b \sin \theta - b \sin 2 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.
Length of Arc of Deltoid
The total length of the arcs of a deltoid constructed within a stator of radius $a$ is given by: :$\LL = \dfrac {16 a} 3$
Equation of Epicycloid
Let a circle $C_1$ of radius $b$ roll without slipping around the outside of a circle $C_2$ of radius $a$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $H$ be the epicycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta}$ :$y = \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta}$
Equation of Nephroid
The point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.
Length of Arc of Nephroid
The total length of the arcs of a nephroid constructed around a stator of radius $a$ is given by: :$\LL = 12 a$
Supremum of Meet Image of Directed Set
Let $\struct {S, \preceq}$ be an up-complete meet semilattice. Let $f: S \times S \to S$ be a mapping such that :$\forall \tuple {x, y} \in S \times S: \map f {x, y} = x \wedge y$ Let $D$ be directed subset of $S \times S$ in Cartesian product $\struct {S \times S, \precsim}$ of $\struct {S, \preceq}$ and $\struct {S, \preceq}$. Then: :$\map \sup {\map {f^\to} D} = \sup \set {x \wedge y: x \in \map {\pr_1^\to} D, y \in \map {\pr_2^\to} D}$ where :$\pr_1$ denotes the first projection on $S \times S$ :$\pr_2$ denotes the second projection on $S \times S$ :$\map {\pr_1^\to} D$ denotes the image of $D$ under $\pr_1$
Meet is Increasing
Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $f: S \times S \to S$ be a mapping such that :$\forall s, t \in S: f\left({s, t}\right) = s \wedge t$ Then: :$f$ is increasing as a mapping from Cartesian product $\left({S\times S, \precsim}\right)$ of $\left({S, \preceq}\right)$ and $\left({S, \preceq}\right)$ into $\left({S, \preceq}\right)$.
Meet Preserves Directed Suprema/Lemma 2
Let $x$ be an element of $S$, $D$ be a directed subset of $S$. Then: :$\paren {\sup D} \wedge x = \sup \set {d \wedge x: d \in D}$
Supremum by Suprema of Directed Set in Cartesian Product
Let $\struct {S, \preceq}$ be an up-complete meet semilattice. Let $\struct {S \times S, \precsim}$ be the Cartesian product of $\struct {S, \preceq}$ and $\struct {S, \preceq}$. Let $D$ be a directed subset of $S \times S$. Then: :$\sup D = \tuple {\map \sup {\map {\pr_1^\to} D}, \map \sup {\map {\pr_2^\to} D} }$
Evolute of Circle is its Center
The evolute of a circle is a single point: its center.
Parametric Equations for Evolute/Formulation 1
Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$. The parametric equations for the evolute of $C$ can be expressed as: :$\begin{cases} X = x - \dfrac {y' \paren {1 + y'^2} } {y''} \\ Y = y + \dfrac {1 + y'^2} {y''} \end{cases}$ where: :$\left({x, y}\right)$ denotes the Cartesian coordinates of a general point on $C$ :$\left({X, Y}\right)$ denotes the Cartesian coordinates of a general point on the evolute of $C$ :$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ {{WRT|Differentiation}} $x$.
Parametric Equations for Evolute/Formulation 2
Let $C$ be a curve expressed as the locus of an equation $\map f {x, y} = 0$. The parametric equations for the evolute of $C$ can be expressed as: :$\begin{cases} X = x - \dfrac {y' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \\ Y = y + \dfrac {x' \paren {x'^2 + y'^2} } {x' y'' - y' x''} \end{cases}$ where: :$\tuple {x, y}$ denotes the Cartesian coordinates of a general point on $C$ :$\tuple {X, Y}$ denotes the Cartesian coordinates of a general point on the evolute of $C$ :$x'$ and $x''$ denote the derivative and second derivative respectively of $x$ {{WRT|Differentiation}} $t$ :$y'$ and $y''$ denote the derivative and second derivative respectively of $y$ {{WRT|Differentiation}} $t$.
Normal to Curve is Tangent to Evolute
Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. Let $P$ be a point on $C$. Let $Q$ be the center of curvature of $C$ at $P$. The normal to $C$ at $P$ is tangent to the evolute $E$ of $C$ at $Q$.
Length of Arc of Evolute equals Difference in Radii of Curvature
Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. The length of arc of the evolute $E$ of $C$ between any two points $Q_1$ and $Q_2$ of $C$ is equal to the difference between the radii of curvature at the corresponding points $P_1$ and $P_2$ of $C$.
Curve is Involute of Evolute
Let $C$ be a curve defined by a real function which is twice differentiable. Let the curvature of $C$ be non-constant. Let $E$ be the evolute $C$. Then the involute of $E$ is $C$.
Evolute of Cycloid is Cycloid
The evolute of a cycloid is another cycloid.
Body behaves as Particle under Gravitation
A body of finite volume under the force of gravity behaves identically to a point mass under the same force.