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Point at infinity

is the point on the line such that . It can also be thought of as the point of intersection of two parallel lines. In 1639, Desargues (1864) became the first to consider the point at infinity (Cremona 1960, p. ix), although Poncelet was the first to systematically employ the point at infinity (Graustein 1930).A point lying on the line at infinity is a point at infinity. In particular, a point with trilinear coordinates is a point at infinity if it satisfiesPoints at infinity therefore do not have exacttrilinear coordinates.Kimberling centers are points at infinity for (the Euler infinity point), 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 674, 680, 681, 688, 690, 696, 698, 700, 702, 704, 706, 708, 710, 712, 714, 716, 718, 720, 722, 724, 726, 730, 732, 734, 736, 740, 742, 744, 746, 752, 754, 758, 760, 766, 768, 772, 776, 778, 780, 782, 784, 786,..


A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple (, , ..., ) consisting of coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his Elements, gave the vague definition of a point as "that which has no part."The basic geometric structures of higher dimensional geometry--the line, plane, space, and hyperspace--are all built up of infinite numbers of points arranged in particular ways.These facts lead to the mathematical pun, "without geometry, life is pointless."The decimal point in a decimal..

Planar distance

For points in the plane, there are at leastdifferent distances. The minimum distance can occur only times, and the maximum distance can occur times. Furthermore, no distance can occur as often astimes.Finally, no set of points in the plane can determine only isosceles triangles.


The point on a line segment dividing it into two segments of equal length. The midpoint of a line segment is easy to locate by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint (Pedoe 1995, p. xii). It is more challenging to locate the midpoint using only a compass (i.e., a Mascheroni construction).For the line segment in the plane determined by and , the midpoint can be calculated as(1)Similarly, for the line segment in space determined by and , the midpoint can be calculated as(2)In a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices (Dunham 1990).In the figure above, the trilinear coordinates of the midpoints of the triangle sides are , , and .The midpoint of a line segment with endpoints and given in trilinear coordinates is , where(3)(4)(5)(left as an exercise in Kimberling..

Line point picking

Consider a line segment of length 1, and pick a point at random between . This point divides the line into line segments of length and . If a set of points are thus picked at random, the resulting distribution of lengths has a uniform distribution on . Similarly, separating the two pieces after each break, the larger piece has uniform distribution on (with mean 3/4), and the smaller piece has uniform distribution on (with mean 1/4).The probability that the line segments resulting from cutting at two points picked at random on a unit line segment determine a triangle is given by 1/4.The probability and distribution functions for the ratio of small to large pieces are given by(1)(2)for . The raw moments are therefore(3)where is the digamma function. The first few are therefore(4)(5)(6)(7)(OEIS A115388 and A115389).The central moments are therefore(8)where is a Pochhammer symbol. The first few are therefore(9)(10)(11)This therefore gives mean,..

Circular point at infinity

The circular points at infinity, also called the isotropic points, are the pair of (complex) points on the line at infinity through which all circles pass. The circular points at infinity belong to the lines with slopes and . In the plane of a triangle, the circular points at infinity are isogonal conjugates of each other.All conics passing through the circular points at infinity are circles.The circular points at infinity are the fixed points of the orthogonalinvolution.Circular points at infinity were first considered by Poncelet in 1813 (Coxeter 1993).

Jung's theorem

Jung's theorem states that the generalized diameter of a compact set in satisfieswhere is the circumradius of (Danzer et al. 1963).This gives the special case that every finite set of points in two dimensions with geometric span has an enclosing circle with radius no greater than (Rademacher and Toeplitz 1957, p. 104).


Given a point , the point which is the antipodal point of is said to be the antipode of .The term antipode is also used in plane geometry. Given a central conic (or circle) and a point lying on it, draw a line passing through and the center of the conic. Then the antipode of is other point lying on the conic through which the line passes.

Point distances

The maximum distance between points in three dimensions can occur no more than times. Also, there exists a fixed number such that no distance determined by a set of points in three-dimensional space occurs more than times. The maximum distance can occur no more than times in four dimensions, where is the floor function.

Antipodal points

Two points are antipodal (i.e., each is the antipode of the other) if they are diametrically opposite. Examples include endpoints of a line segment, or poles of a sphere. Given a point on a sphere with latitude and longitude , the antipodal point has latitude and longitude (where the sign is taken so that the result is between and ).

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