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The convexity coefficient chi(D) of a region D is the probability that the line segment connecting two random points in D is contained entirely within D. For a convex region, ...

The quadrifolium is the 4-petalled rose curve having n=2. It has polar equation r=asin(2theta) (1) and Cartesian equation (x^2+y^2)^3=4a^2x^2y^2. (2) The area of the ...

Finding the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which the it must be filled for it to be one quarter full amounts to plugging ...

The "trefoil" curve is the name given by Cundy and Rollett (1989, p. 72) to the quartic plane curve given by the equation x^4+x^2y^2+y^4=x(x^2-y^2). (1) As such, it is a ...

The surface area of a spherical segment. Call the radius of the sphere R, the upper and lower radii b and a, respectively, and the height of the spherical segment h. The zone ...

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates theta and phi from uniform distributions theta in [0,2pi) and phi in ...

Let A be the area of a simply closed lattice polygon. Let B denote the number of lattice points on the polygon edges and I the number of points in the interior of the ...

Construct a square equal in area to a circle using only a straightedge and compass. This was one of the three geometric problems of antiquity, and was perhaps first attempted ...

Any bounded planar region with positive area >A placed in any position of the unit square lattice can be translated so that the number of lattice points inside the region ...

A regular two-dimensional arrangement of squares separated by vertical and horizontal "canals." Looking at the grid produces the illusion of gray spots in the white area ...