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How do you calculate cross ratio?

How do you calculate cross ratio?

Given a line (e) and four points A, B, C, and D on it, the cross ratio of these four points is defined by the equation: (ABCD) = ((a-c)/(b-c))/((a-d)/(b-d)). The right side of this equation is defined as the cross ratio of four real numbers.

What do you mean by cross ratio?

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line.

What is the dual of Pappus?

The projectively dual of the Pappus theorem is the same as the original theorem. This is equivalent with the dual producing the same figure as the original. Pappus theorem deals with an hexagon ABCDEF inscribed in the degenerate conic of P2 (projective plane) defined by two lines (a) and (d).

What is the Pappus line?

In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point.

What is cross ratio in complex analysis?

Complex cross ratio The complex cross ratio of four points in the (complex) plane is defined to be: (ABCD) = ((A-C)/(B-C))/((A-D)/(B-D)), where points are identified with complex numbers. [1] (ABCD) is real if and only if the points are all four on a circle or line.

What is cross ratio in bilinear transformation?

Theorem. A bilinear transformation preserves cross-ratio of four points i.e. (w1 – w2)(w3 – w4)

What was the size of Jesus cross?

He determined the Jesus cross weighed 165 pounds, was three or four meters high, with a cross beam two meters wide.

What dimensions should a cross be?

Wood Cross Dimensions A good guideline is to look for one 9′-12′ 4×4 and one 6′ 4×4 to construct the cross. If the cross is going to be mounted in the ground outside the vertical beam will need to be longer because it has to be sunk into the ground approximately three feet.

How did Pappus prove his theorem?

Pappus of Alexandria (fl. ad 320) proved that the three points (x, y, z) formed by intersecting the six lines that connect two sets of three collinear points (A, B, C; and D, E, F) are also collinear. Encyclopædia Britannica, Inc.

What is cross ratio in projective geometry?

cross ratio, in projective geometry, ratio that is of fundamental importance in characterizing projections. In a projection of one line onto another from a central point (see Figure), the double ratio of lengths on the first line (AC/AD)/(BC/BD) is equal to the corresponding ratio on the other line.

What is the Pappus configuration in geometry?

The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus’s theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of ABC and abc.

What are the equivalent statements of Pappus’s theorem?

In addition to the above characterizations of Pappus’s theorem and its dual, the following are equivalent statements: If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.

What is the origin of cross ratio?

The theorem has been established by Pappus in the seventh book of his Mathematical Collections. It was further developed by Desargues starting with 1639 [Wells, p. 41]. As defined in (1), the cross ratio is meaningful for any four points not necessarily collinear.

How many different cross-ratios can four points have in 4?

Four points can be ordered in 4! = 4 × 3 × 2 × 1 = 24 ways, but there are only six ways for partitioning them into two unordered pairs. Thus, four points can have only six different cross-ratios, which are related as: See Anharmonic group below.