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A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle.
The perimeter of a stadium is calculated by the formula = (+) where a is the length of the straight sides and r is the radius of the semicircles. With the same parameters, the area of the stadium is A = π r 2 + 2 r a = r ( π r + 2 a ) {\displaystyle A=\pi r^{2}+2ra=r(\pi r+2a)} .
In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI of book 1 of Newton's Principia, and used it to show that the position of a planet moving in an orbit is not an algebraic function of time.
Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96 / 2 − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.
In a projective plane a set Ω of points is called an oval, if: Any line l meets Ω in at most two points, and; For any point P ∈ Ω there exists exactly one tangent line t through P, i.e., t ∩ Ω = {P}. For finite planes (i.e. the set of points is finite) there is a more convenient characterization: [2]
Then the Cartesian oval is the locus of points S satisfying d(P, S) + m d(Q, S) = a. The two ovals formed by the four equations d( P , S ) + m d( Q , S ) = ± a and d( P , S ) − m d( Q , S ) = ± a are closely related; together they form a quartic plane curve called the ovals of Descartes .
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.