Spread Rational trigonometry
1 spread
1.1 calculating spread
1.1.1 trigonometric
1.1.2 vector/slope (two-variable)
1.1.3 cartesian (three-variable)
1.2 spread compared angle
1.3 turn , coturn
1.4 twist
spread
suppose l1 , 12 intersect @ point a. let c foot of perpendicular b 12. spread s = q/r.
spread gives 1 measure separation of 2 lines single dimensionless number in range [0,1] (from parallel perpendicular) euclidean geometry. replaces concept of angle has several differences angle, discussed in section below. spread can have several interpretations.
trigonometric (most elementary): sine ratio quadrances in right triangle (left) , therefore equivalent square of sine of angle. viewing side ac part of unit diameter of circle , considering similar triangles (right), spread measured exterior segment, equal 1 half times (1 minus cosine of central angle), twice angle @ a, , haversine.
vector: rational function of relative directions (or slopes) of pair of lines meet.
cartesian: rational function of 3 co-ordinates used ascribe 2 vectors.
linear algebra (from dot product): normalized rational function: square of determinant of 2 vectors (or pair of intersecting lines) divided product of quadrances.
calculating spread
trigonometric
suppose 2 lines, l1 , l2, intersect @ point shown @ right. choose point b ≠ on l1 , let c foot of perpendicular b l2. spread s is
s
(
ℓ
1
,
ℓ
2
)
=
q
(
b
,
c
)
q
(
a
,
b
)
=
q
r
.
{\displaystyle s(\ell _{1},\ell _{2})={\frac {q(b,c)}{q(a,b)}}={\frac {q}{r}}.}
vector/slope (two-variable)
like angle, spread depends on relative slopes of 2 lines (constant terms being eliminated) , invariant under translation (i.e. preserved when lines moved keeping parallel themselves). given 2 lines equations are
a
1
x
+
b
1
y
=
constant
and
a
2
x
+
b
2
y
=
constant
{\displaystyle a_{1}x+b_{1}y={\text{constant}}\qquad {\text{and}}\qquad a_{2}x+b_{2}y={\text{constant}}}
we may rewrite them 2 lines meet @ origin (0, 0) equations
a
1
x
+
b
1
y
=
0
and
a
2
x
+
b
2
y
=
0
{\displaystyle a_{1}x+b_{1}y=0\qquad {\text{and}}\qquad a_{2}x+b_{2}y=0}
in position point (−b1, a1) satisfies first equation , (−b2, a2) satisfies second , 3 points (0, 0), (−b1, a1) , (−b2, a2) forming spread give 3 quadrances:
q
1
=
(
b
1
2
+
a
1
2
)
,
q
2
=
(
b
2
2
+
a
2
2
)
,
q
3
=
(
b
1
−
b
2
)
2
+
(
a
1
−
a
2
)
2
{\displaystyle {\begin{aligned}q_{1}&=\left(b_{1}^{2}+a_{1}^{2}\right),\\q_{2}&=\left(b_{2}^{2}+a_{2}^{2}\right),\\q_{3}&=\left(b_{1}-b_{2}\right)^{2}+\left(a_{1}-a_{2}\right)^{2}\end{aligned}}}
the cross law – see below – in terms of spread is
1
−
s
=
(
q
1
+
q
2
−
q
3
)
2
4
q
1
q
2
.
{\displaystyle 1-s={\frac {(q_{1}+q_{2}-q_{3})^{2}}{4q_{1}q_{2}}}.}
which becomes:
1
−
s
=
(
a
1
2
+
a
2
2
+
b
1
2
+
b
2
2
−
(
b
1
−
b
2
)
2
−
(
a
1
−
a
2
)
2
)
2
4
(
a
1
2
+
b
1
2
)
(
a
2
2
+
b
2
2
)
.
{\displaystyle 1-s={\frac {\left(a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2}-(b_{1}-b_{2})^{2}-(a_{1}-a_{2})^{2}\right)^{2}}{4\left(a_{1}^{2}+b_{1}^{2}\right)\left(a_{2}^{2}+b_{2}^{2}\right)}}.}
this simplifies, in numerator, (2a1a2 + 2b1b2), giving:
1
−
s
=
(
a
1
a
2
+
b
1
b
2
)
2
(
a
1
2
+
b
1
2
)
(
a
2
2
+
b
2
2
)
.
{\displaystyle 1-s={\frac {\left(a_{1}a_{2}+b_{1}b_{2}\right)^{2}}{\left(a_{1}^{2}+b_{1}^{2}\right)\left(a_{2}^{2}+b_{2}^{2}\right)}}.}
(note: 1 − s expression cross, square of cosine of either angle between pair of lines or vectors, gives name cross law.)
then, using brahmagupta–fibonacci identity
(
a
2
b
1
−
a
1
b
2
)
2
+
(
a
1
a
2
+
b
1
b
2
)
2
=
(
a
1
2
+
b
1
2
)
(
a
2
2
+
b
2
2
)
,
{\displaystyle \left(a_{2}b_{1}-a_{1}b_{2}\right)^{2}+\left(a_{1}a_{2}+b_{1}b_{2}\right)^{2}=\left(a_{1}^{2}+b_{1}^{2}\right)\left(a_{2}^{2}+b_{2}^{2}\right),}
the standard expression spread in terms of slopes (or directions) of 2 lines becomes
s
=
(
a
1
b
2
−
a
2
b
1
)
2
(
a
1
2
+
b
1
2
)
(
a
2
2
+
b
2
2
)
.
{\displaystyle s={\frac {\left(a_{1}b_{2}-a_{2}b_{1}\right)^{2}}{\left(a_{1}^{2}+b_{1}^{2}\right)\left(a_{2}^{2}+b_{2}^{2}\right)}}.}
in form (and in cartesian equivalent follows) spread ratio of square of determinant of 2 vectors (numerator) product of quadrances (denominator)
cartesian (three-variable)
this replaces (−b1, a1) (x1, y1), (−b2, a2) (x2, y2) , origin (0, 0) (as point of intersection of 2 lines) (x3, y3) in previous result:
s
=
(
(
y
1
−
y
3
)
(
x
2
−
x
3
)
−
(
y
2
−
y
3
)
(
x
1
−
x
3
)
)
2
(
(
y
1
−
y
3
)
2
+
(
x
1
−
x
3
)
2
)
(
(
y
2
−
y
3
)
2
+
(
x
2
−
x
3
)
2
)
.
{\displaystyle s={\frac {{\bigl (}(y_{1}-y_{3})(x_{2}-x_{3})-(y_{2}-y_{3})(x_{1}-x_{3}){\bigr )}^{2}}{{\bigl (}(y_{1}-y_{3})^{2}+(x_{1}-x_{3})^{2}{\bigr )}{\bigl (}(y_{2}-y_{3})^{2}+(x_{2}-x_{3})^{2}{\bigr )}}}.}
spread compared angle
the spread of 2 lines can measured in 4 equivalent positions.
unlike angle, can define relationship between rays emanating point, circular measure parametrization, , pair of lines can considered 4 pairs of rays, forming 4 angles, spread more fundamental in rational trigonometry, describing 2 lines single measure of rational function (see above). being equivalent square of sine of corresponding angle θ (and haversine of chord-based double-angle Δ = 2θ), spread of both angle , supplementary angle equal.
spread not proportional, however, separation between lines angle be; spreads of 0, 1/4, 1/2, 3/4, , 1 corresponding unevenly spaced angles 0°, 30°, 45°, 60° , 90°.
instead, (recalling supplementary property) 2 equal, co-terminal spreads determine third spread, value solution of triple spread formula triangle (or 3 concurrent lines) spreads of s, s , r:
(
2
s
+
r
)
2
=
2
(
2
s
2
+
r
2
)
+
4
s
2
r
4
s
2
+
4
s
r
+
r
2
=
4
s
2
+
2
r
2
+
4
s
2
r
{\displaystyle {\begin{aligned}(2s+r)^{2}&=2\left(2s^{2}+r^{2}\right)+4s^{2}r\\4s^{2}+4sr+r^{2}&=4s^{2}+2r^{2}+4s^{2}r\end{aligned}}}
giving quadratic polynomial (in s):
r
2
+
4
s
2
r
−
4
s
r
=
0
r
2
−
4
s
(
1
−
s
)
r
=
0
{\displaystyle {\begin{aligned}r^{2}+4s^{2}r-4sr&=0\\r^{2}-4s(1-s)r&=0\end{aligned}}}
and solutions
r
=
0
(
trivial
)
or
r
=
4
s
(
1
−
s
)
{\displaystyle r=0\quad ({\text{trivial}})\qquad {\text{or}}\qquad r=4s(1-s)}
this equivalent trigonometric identity :
sin
2
(
2
θ
)
=
4
sin
2
θ
(
1
−
sin
2
θ
)
{\displaystyle \sin ^{2}(2\theta )=4\sin ^{2}\theta \left(1-\sin ^{2}\theta \right)}
of angles θ, θ , 180° − 2θ of triangle, using
s
2
(
s
)
=
s
2
(
sin
2
θ
)
=
sin
2
(
2
θ
)
=
r
(
s
)
{\displaystyle s_{2}(s)=s_{2}\left(\sin ^{2}\theta \right)=\sin ^{2}(2\theta )=r(s)}
to denote second spread polynomial in s.
tripling spreads likewise involves triangle (or 3 concurrent lines) 1 spread of r (the previous solution), 1 spread of s , obtaining third spread polynomial, t in s. turns out be:
s
3
(
s
)
=
s
(
3
−
4
s
)
2
=
t
(
s
)
{\displaystyle s_{3}(s)=s(3-4s)^{2}=t(s)}
further multiples of basic spread of lines can generated continuing process using triple spread formula.
every multiple of spread rational rational, converse not apply. example, half-angle formula, 2 lines meeting @ 15° (or 165°) angle have spread of:
hav
(
30
∘
)
=
sin
2
(
30
∘
2
)
=
1
−
cos
30
∘
2
=
1
−
3
2
2
=
2
−
3
4
≈
0.0667.
{\displaystyle \operatorname {hav} \left(30^{\circ }\right)=\sin ^{2}\left({\frac {30^{\circ }}{2}}\right)={\frac {1-\cos 30^{\circ }}{2}}={\frac {1-{\frac {\sqrt {3}}{2}}}{2}}={\frac {2-{\sqrt {3}}}{4}}\approx 0.0667.}
and exists algebraic extension of rational numbers.
turn , coturn
twist
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