Chapter I
Transformation by Reciprocal Radii
Transformation by reciprocal radii refers in general to three-dimensional space. I expose this transformation in reference to the plane, or rather to two overlapping planes.
Each point on one of the two planes corresponds to another point on the other plane, and vice versa. Overlapping points are defined as fixed, that is, they correspond to themselves. Fixed are the points of the circumference with respect to which the transformation is carried out.
An important exception is the following: all the points at infinity (i.e., the directions of the infinite number of straight lines) correspond to only one point, the center of the circle with respect to which the transformation is carried out, and vice versa.
The geometric inversion (by reciprocal radii) is a quadratic or Cremona transformation and has the following properties:
- With respect to a circle, it transforms arcs into arcs.
- Transforms straight lines into circles passing through the center of inversion (
O). - The straight line passing through
Otransforms into itself.
The inversion is an isogonal or conformal correspondence, which means it preserves the angles but reverses their orientation.
The inversion extends to the 3rd coordinate (sphere) with the same properties:
- Spheres are transformed into spheres.
- Planes into spheres passing through the center of inversion, and vice versa.
To the plane at infinity, that is to all directions of space, corresponds the center O′ of the sphere with respect to which the inversion is carried out. We will treat the transformation referred to the plane, for simplicity and clarity. Each internal point of the circle of inversion corresponds to an external one, and vice versa.
In Table I, we considered two circles (however considered overlapping):
- If we overlap the two circles, we will have, in the same figure, the internal curved tangent and the external straight tangent, which correspond to each other.
- The two overlapping points of contact constitute a single fixed point.
At the left in Table II, we have the geometric procedure of inversion, to obtain the internal point of the circle that corresponds to an external point and vice versa.
Given a circle with a radius (e.g., 1 meter), we consider point 2 (2m away from the center of the circle) and draw from 2 the two tangents to the circle, passing through the two contact points a and b. Now we consider the point where the line segment joining a and b intersects the line segment joining 2 with the center of the circle: the point of intersection is 1/2 (half a meter), which is the multiplicative inverse (or reciprocal) of 2 (hence the name inversion or reciprocal radii).
To the generic external point m will correspond the internal point 1/m and vice versa. If it is a point at infinity, then from it are drawn parallel tangents that touch the circle at the endpoints of a diameter of the given circle. To this generic point at infinity will correspond the center of the circle, that is, as already mentioned, to each point at infinity (direction) corresponds exactly one point, the center of the circle of inversion.
To find the center N of a circular arc OP, an arc that corresponds to an external segment of the straight line C, we consider the small figure at the right in Table II where the external non-dashed line segment of C corresponds to the arc OP passing through O and through the fixed point P.
The searched center N is located at the intersection of the extension of the circle’s diameter with the perpendiculars on the midpoint of the chord OP (see Table II).
To the dashed straight Euclidean segment inside the circle of inversion corresponds the completion of the non-Euclidean arc external to the circle (also see Table XI).
Let’s consider Table IV; to each curve in the upper figure corresponds a straight line in the lower figure. The two figures, as already mentioned, should be thought of as overlapping. The upper figure represents the non-Euclidean space; the lower figure represents the Euclidean space (where Euclid's 5th postulate is valid).
- To the straight tangents
ab,bc,cdof the Euclidean space (lower figure) correspond the curved tangentsab,bc,cdof the space with variable curvature (upper figure). - To the straight Euclidean parallels correspond the curved non-Euclidean parallels.
- The angles intersecting the Euclidean lines and the corresponding non-Euclidean lines are identical.
The reversible formulas for the transformation from the classical exospherical cosmos into the endospherical one are:
x = r² x₁ / (x₁² + y₁²)
y = r² y₁ / (x₁² + y₁²)
where x₁ and y₁ are the inverse coordinates of x and y.
Projectivity
Projectivity (also known as homography or projective transformation) is a bijective algebraic correspondence between S₁ and S′₁ or a bijective and continuous correspondence that preserves cross-ratios.
Involution is a special case of projectivity between two forms of the first kind, where any two elements always correspond in a double way.
The two elements are said to be conjugated in the involution, which has two fixed or double points in each of which two conjugated elements coincide.
A conic establishes a correspondence, subject to the conic, between the points and lines of a plane: this correspondence is called polarity. An involutive correlation between two overlapping planes is a plane polarity.
If a point P and a plane p have a reciprocal correspondence in the polarity, then they are said to be the pole and polar of each other. If the second point lies on the polar of the first, the first will lie on the polar of the second: the two points are said to be conjugated or reciprocal in polarity. A point is called self-conjugate if it lies on its own polar.
A polarity is represented by equations of the form:
pu = a₁₁x + a₁₂y + a₁₃z
pv = a₂₁x + a₂₂y + a₂₃z
pw = a₃₁x + a₃₂y + a₃₃z
The condition for two points P(x, y, z) and Q(x′, y′, z′) to be conjugated is:
vx′ + vy′ + wz′ = 0