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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 ...

Picking two independent sets of points x and y from a unit uniform distribution and placing them at coordinates (x,y) gives points uniformly distributed over the unit square. ...

The equation of a line ax+by+c=0 in slope-intercept form is given by y=-a/bx-c/b, (1) so the line has slope -a/b. Now consider the distance from a point (x_0,y_0) to the ...

Taking the locus of midpoints from a fixed point to a circle of radius r results in a circle of radius r/2. This follows trivially from r(theta) = [-x; 0]+1/2([rcostheta; ...

Given a "good" graph G (i.e., one for which all intersecting graph edges intersect in a single point and arise from four distinct graph vertices), the crossing number is the ...

Let a line in three dimensions be specified by two points x_1=(x_1,y_1,z_1) and x_2=(x_2,y_2,z_2) lying on it, so a vector along the line is given by v=[x_1+(x_2-x_1)t; ...

Number Theory

If a univariate real function f(x) has a single critical point and that point is a local maximum, then f(x) has its global maximum there (Wagon 1991, p. 87). The test breaks ...

The second de Villiers point is the perspector of the reference triangle and the excenter analog of the BCI triangle, which is Kimberling center X_(1128) has triangle center ...

An odd alternating permutation number, more commonly called an Euler number or secant number.