ssrs barcode font Non-Euclidean Geometry in Objective-C

Generator QR in Objective-C Non-Euclidean Geometry

CHAPTER 19 Non-Euclidean Geometry
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The reason why hyperbolic space is the one that satisfies Postulate 5b can be seen in Fig. 19-7. Because of all the frilly waves4 space through which a line can travel, there is more than one line through point P of that will never cross AB. Note that these do not have the usual property of parallel lines; they are not everywhere equidistant.
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Fig. 19-7
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In hyperbolic space Planes are negatively curved and infinite in area. Lines are infinite in length. The angles of a triangle add up to less than 180 . One such example is illustrated in Fig. 19-8.
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Fig. 19-8
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Just as with elliptic space, the larger a triangle is, the further its angle sum will be from 180 . A tiny triangle will have angles that add up to just less than 180 . The largest triangles in hyperbolic space have angle sums of almost zero. Because of this, there are no similar triangles of different sizes in hyperbolic space. Finally, a circle has more circumference and encompasses more area in hyperbolic space than it would in the euclidean plane. Thus, a circle with radius r in hyperbolic space has circumference C 2pr and area A pr2.
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Formulas for Reference
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Angle Formulas 1. 2. 3. 4. 5. 6. Complement of a Supplement of a Sum of measures of angles of a triangle Sum of measures of angles of a quadrilateral Sum of measures of exterior angles of an n-gon Sum of measures of interior angles of an n-gon 1. 2. 3. 4. 5. 6. c s S S S S 90 a 180 a 180 360 360 180 (n 2) 180 (n 2) n 360 n a
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1 2a 1 2a 1 2 (a 1 2 (a
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7. Measure of each interior angle of an equiangular or regular n-gon 8. Measure of each exterior angle of an equiangular or regular n-gon 9. Measure of central /O intercepting an arc of a 10. Measure of inscribed /A intercepting an arc of a 11. Measure of /A formed by a tangent and a chord and intercepting an arc of a 12. Measure of /A formed by two intersecting chords and intercepting arcs of a and b 13. Measure of /A formed by two intersecting tangents, two intersecting secants, or an intersecting tangent and secant and intercepting arcs of a and b 14. Measure of /A inscribed in a semicircle s 15. Opposite /A and B of an inscribed quadrilateral
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7. S 8. S
9. m/O 10. m/ A 11. m/A 12. m/A 13. m/A
b) b)
14. m/A 15. m/A
90 180
Formulas for Reference
Area Formulas 1. Area of a rectangle 2. Area of a square 3. Area of a parallelogram 4. Area of a triangle 5. Area of a trapezoid 6. Area of an equilateral triangle 7. Area of a rhombus 8. Area of a regular polygon 9. Area of a circle 10. Area of a sector 11. Area of a minor segment 1. A 2. A 3. A 4. A 5. A 6. A 7. A 8. A 9. A 10. A 11. A bh s2, bh,
1 2 bh, 1 2 h(b
A A A
1 2 2d
ab sin C
1 2 ab
sin C
1 2 3 h 23
1 2 4 s 23, 1 n 2 dd 1 2 pr
pr 2, n (pr 2) 360 area of sector
1 2 4 pd
area of triangle
Circle Intersection Formulas 1. 2. 3.
Intersecting Chords ab cd
Intersecting Tangent and Secant t 2 s se e, t t
Intersecting Secants se s e
Right-Triangle Formulas 1. Pythagorean Theorem 1. c2 a2 b2
Leg opposite 30 angle Leg opposite 45 angle Leg opposite 60 angle
2. b b a
1 2c 1 2 c 22, 1 2 c 23,
a b 23
Altitude of equilateral triangle Side of equilateral triangle
3. h s
1 2 s 23 2 3 h 23
(contd.)
Formulas for Reference
Right-Triangle Formulas 4. Side of square Diagonal of square 4. s d p h c a c b
1 2 d 22
s 22 h 2 q, h a 2 p, a b 2 q, b 2pq 2pc 2qc
Altitude to hypotenuse Leg of right triangle
pq, h pc, a qc, b
Coordinate-Geometry Formulas 1. Midpoint M Distance P P2 1 Slope of P1 P2 2. Slopes of parallels, L1 and L2 Slopes of perpendiculars, L1 and L 3. Equation of L1, parallel to x-axis Equation of L2, parallel to y-axis
1. xM d m
x1 2
, yM
y1 2 (y2 y ,m x
y2 y1)2 tan i
2(x2 x1)2 y2 y1 x2 x1 , m
2. Same slope, m mm 1 1 m m,m
3. y x
Equation of L1 with slope m and y-intercept b Equation of L2 with slope m passing through the origin Equation of L1 with x-intercept a and y-intercept b Equation of L3 with slope m and passing through (x1, y1) Equation of circle with center at origin and radius r
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