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Investigation: Interpret and systematize the key formulas of universal hyperbolic geometry. Universal hyperbolic geometry Write down all of the results of Universal Hyperbolic Geometry. Organize them and interpret them in terms of
Understand and interpret
Exercises Formulate analogues of the deep ideas behind geometry and trigonometry, such as: A right triangle is half a rectangle. A triangle is the sum of two right triangles. Four times a right triangle is the difference of two squares. Literature
Duality In hyperbolic geometry, the quadrance {$q$} (between points) and spread {$S$} (between lines) are dual. Each is equal to the crossratio. Quadrance Spread Duality Theorem. If {$a_1=A_1^{\perp}$} and {$a_2=A_2^{\perp}$}, then {$R(a_1,b_2;a_2,b_1)=q(a_1,a_2)=S(A_1,A_2)$}. Where {$a_1,a_2$} are points within a circle and {$b_1,b_2$} are points on their respective polars. Foundations In the plane, two vectors indicate the same line if the area they span is zero. {$x_1:y_1 = x_2:y_2 \iff \frac{x_1}{y_1} = \frac{x_2}{y_2} \iff x_1y_2  x_2y_1 = 0$} In threedimensional space, two vectors indicate the same line if, in each of the three planes, the area they span is zero. {$x_1:y_1:z_1 = x_2:y_2:z_2 \iff \frac{x_1}{y_1} = \frac{x_2}{y_2} \wedge \frac{x_1}{z_1} = \frac{x_2}{z_2} \wedge \frac{y_1}{z_1} = \frac{y_2}{z_2} \iff x_1y_2  x_2y_1 = x_1z_2  x_2z_1 = y_1z_2  y_2z_1 = 0$} Given point a = [x:y:z] and line L = (l:m:n). a is on L (L passes through a) when {$xl+my=nz$} or when {$xl+mynz=0$} or when {$a\circ L=0$}. a and L are dual ({$a^{\perp}=L$} and {$a=L^{\perp}$}) when [x:y:z]=[l:m:n] or (x:y:z)=(l:m:n). a is null when it lies on its dual line, thus when {$x^2+y^2=z^2$}, (when {$a\circ a^{\perp}=0$}), and likewise, L is null when it passes through its dual point. Points {$a_1=[x_1:y_1:z_1]$} and {$a_2=[x_2:y_2:z_2]$} are perpendicular when {$x_1x_2+y_1y_2=z_1z_2$} or when {$x_1x_2+y_1y_2z_1z_2=0$} or when {$a_1\circ a_2=0$}. Similarly with lines. Overview We have laws for:
The squares of the six trigonometric functions find expression as the six possible crossratios: q = sin2, 1q = cos2, 1/q = csc2, 1/(1q) = sec2, q/(1q) = tan2, (1q)/q = cot2. Similarly, the six hyperbolic functions express the quadrance. Symmetric formulas Spread law. {$\frac{S_1}{q_1}=\frac{S_2}{q_2}=\frac{S_3}{q_3}$} Triple quad formula. If {$a_1, a_2, a_3$} collinear, then {$(q_1+q_2+q_3)^2=2(q_1^2+q_2^2+q_3^2)+4q_1q_2q_3$}, which is {$p_1^2=2p_2+4e_3$}. Triple spread formula. If {$A_1, A_2, A_3$} concurrent, then {$(S_1+S_2+S_3)^2=2(S_1^2+S_2^2+S_3^2)+4S_1S_2S_3$}, which is {$p_1^2=2p_2+4e_3$}. These formula can be rewritten: {$q_1q_2q_3=(2(q_1^2+q_2^2+q_3^2)(q_1+q_2+q_3)^2)/4$}, in other words, {$e_3=(2p_2p_1^2)/4$} Nonsymmetric formulas Pythagoras's theorem. If {$a_1a_3\perp a_2a_3$} then {$q_3=q_1+q_2q_1q_2$}. More elegantly: {$(1q_3)=(1q_1)(1q_2)$}. Pythagoras's dual theorem. If {$A_1A_3\perp A_2A_3$} then {$S_3=S_1+S_2S_1S_2$}. More elegantly: {$(1S_3)=(1S_1)(1S_2)$}. Alternatively, {$\textrm{cos}\,\theta_3=\textrm{cos}\,\theta_1\textrm{cos}\,\theta_2$}. {$(q_1q_2S_3(q_1+q_2+q_3)+2)^2=4(1q_1)(1q_2)(1q_3)$} Cross law. {$(q_1q_2S_3(q_1+q_2+q_3)+2)^2=4(1q_1)(1q_2)(1q_3)$} {$q_1q_2S_3=q_1S_2q_3=S_1q_2q_3= q_1+q_2+q_3 2 \pm 2\sqrt{(1q_1)(1q_2)(1q_3)}$} Cross dual law. {$(S_1S_2q_3(S_1+S_2+S_3)+2)^2=4(1S_1)(1S_2)(1S_3)$} {$S_1S_2q_3=S_1q_2S_3=q_1S_2S_3= S_1+S_2+S_3 2 \pm 2\sqrt{(1S_1)(1S_2)(1S_3)}$} Note that {$\sqrt{1S_i}=\textrm{cos}\,\theta_i$} Theorems The altitudes of a triangle meet at a point (the orthocenter). Affine geometry Lines are parallel:
Affine combinations Vector is the relation between parallel lines. Projective geometry Space is divided into "infinity" and "finity". Conformal geometry Lines are perpendicular: Pythagorean theorem, inner product Quadratic interpolations: law of cosines A2  2ABcos(theta) + B2 = C2 interpolation of (AB)2 = C2, A2 + B2 = C2, (A+B)2 = C2. Consider triangles more generally. Or consider the function: A2 + B2  C2 which is 0 when A and B are perpendicular, and nonzero 2ABcos(theta) otherwise. Inner product defines the equation of a line and whether points lie on it. (a1, a2, a3)(x1, x2, 1) = 0. Symplectic geometry Oriented area, volume given by determinant.
What if we interpret the line as ax + by + cz = 0 where z=1 ? Why is the s1s2s3 term of the third power and not the second power? Notes
N J Wildberger. Chromogeometry Symmetric bilinear forms given vectors {$A_1=[x_1,y_1]$} and {$A_2=[x_2,y_2]$}
Definitions
{$$s(A_1A_2, B_1B_2) ≡ 1 − \frac{((A_2 − A_1) · (B_2 − B_1))^2}{Q(A_1, A_2) Q(B_1, B_2)}$$}. This is {$\textrm{sin}^2\theta$}. It is independent of the choice of points lying on the two lines. Two nonnull lines are perpendicular precisely when the spread between them is 1. Five main laws of planar rational trigonometry
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