Fast solutions for complex problems

How is cross ratio calculated?

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.

How do you find the cross ratio in a complex analysis?

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: Invariance of Cross-Ratio Under Bilinear Transformation Under a linear fractional transformation w = az + b cz + d , the cross-ratio R(z1,z2,z3,z4) becomes R(w1,w2,w3,w4) but is invariant (unchanged).

Does a bilinear transformation preserves cross ratio of four points?

cross ratio remains invariant under the bilinear transformation.

What is the ratio of collinear points?

If four collinear points are represented in homogeneous coordinates by vectors a, b, c, d such that c = a + b and d = ka + b, then their cross-ratio is k.

Is projective geometry Euclidean?

In Euclidean geometry, the sides of objects have lengths, intersecting lines determine angles between them, and two lines are said to be parallel if they lie in the same plane and never meet. Euclidean geometry is actually a subset of what is known as projective geometry.

Is log Z analytic?

Answer: The function Log(z) is analytic except when z is a negative real number or 0.

How do you calculate bilinear transformation?

Find the bilinear transformation which maps the points 2, I, -2 into points 1, I, -1 by using cross ratio property. ∴w=a2−ai−a2−ai∴w=−a(−2+i)−a(2+i)∴w=i−2z+iis the bilinear transformation.

How do you show that points are collinear?

Three or more points are said to be collinear if they all lie on the same straight line. If A, B and C are collinear then. If you want to show that three points are collinear, choose two line segments, for example.

What is a distance ratio?

Basically, it means if two objects have their speeds in the ratio a:b, Distance covered by the two objects in same time will be in the ratio a:b; Time taken by the two objects to cover same distance will be in the ratio b:a. Let us apply this concept to some examples given below. Solved Examples.

Is projective geometry useful?

Projective geometry is also useful in avoiding edge cases of particular configurations, particularly the case of parallel lines (as in projective geometry, there are no parallel lines).

How do you interpret projective geometry?

Projective geometry can be thought of as the collection of all lines through the origin in three-dimensional space. That is, each point of projective geometry is actually a line through the origin in three-dimensional space. The distance between two points can be thought of as the angle between the corresponding lines.

Which is the correct definition of the 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. Given four points A, B, C and D on a line, their cross ratio is defined as where an orientation of the line determines the sign…

What are the proportions of a Christian cross?

In addition, the left and right sides of the cross piece and the top section of the vertical bar are all the same size. Other Christian crosses, such as those from early eras or from Eastern Orthodox sects, use a cross in which the crosspiece and vertical bar meet in the center of both lines.

How is the cross ratio used in non Euclidean geometry?

Arthur Cayley and Felix Klein found an application of the cross-ratio to non-Euclidean geometry. Given a nonsingular conic C in the real projective plane, its stabilizer GC in the projective group G = PGL (3, R) acts transitively on the points in the interior of C. However, there is an invariant for the action of GC on pairs of points.

When does a fourth point make the cross ratio Minus One?

Given three points on a line, a fourth point that makes the cross ratio equal to minus one is called the projective harmonic conjugate. In 1847 Carl von Staudt called the construction of the fourth point a throw ( Wurf ), and used the construction to exhibit arithmetic implicit in geometry.