Chapter 4: Geometry

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5. Re ec tion in a line with equation Ü · Ý · ¼: ´Ü Ýµ ½ ¾ ¾ ¾ ´Ü ¾ · ¾ ¾ ¾ ¾ Ýµ ´¾ ¾µ (4.2.5) 6. Re ec tion in a line going through ´Ü ¼ Ý ¼ µ and making an angle « with the Ü-axis: ´Ü Ýµ ´Ü ¼ Ý ¼ µ· Ó× ¾« ×Ò ¾« ´Ü Ü ×Ò ¾« Ó× ¾« ¼ Ý Ý ¼ µ (4.2.6) 7. Glide-re ection in a line Ä with displacement : Apply rst a re ection in Ä, then a translation by a vector of length in the direction of Ä, that is, by the vector ½ ´¦ ¾ §µ (4.2.7) · ¾ if Ä has equation Ü · Ý · ¼. 4.2.2 FORMULAE FOR SYMMETRIES: HOMOGENEOUS COORDINATES All isometries of the plane can be expressed in homogeneous coordinates in terms of multiplication by a matrix. This fact is useful in implementing these transformations on a computer. It also means that the successive application of transformations reduces to matrix multiplication. The corresponding matrices are as follows: 1. Translation by ´Ü ¼ Ý ¼ µ: Ì´Ü¼Ý ¼µ ¾ ¿ ½ ¼ Ü ¼ ¼ ½ Ý ¼ (4.2.8) ¼ ¼ ½ 2. Rotation through « around the origin: Ê « ¾ ¿ Ó× « ×Ò « ¼ ×Ò « Ó× « ¼ (4.2.9) ¼ ¼ ½ 3. Re ec tion in a line going through the origin and making an angle « with the Ü-axis: ¾ ¿ Ó× ¾« ×Ò ¾« ¼ Å « ×Ò ¾« Ó× ¾« ¼ (4.2.10) ¼ ¼ ½ From this one can deduce all other transformations. © 2003 by CRC Press LLC
EXAMPLE To nd the matrix for a rotation through « around an arbitrary point È ´Ü ¼Ý ¼µ, we apply a translation by ´Ü ¼Ý ¼µ to move È to the origin, a rotation through « around the origin, and then a translation by ´Ü ¼Ý ¼µ: Ì´Ü¼ Ý¼µÊ «Ì ´Ü¼Ý¼µ (notice the order of the multiplication). ¾ Ó× « ×Ò « Ü ¼ Ü ¼ Ó× « · Ý ¼ ×Ò « ×Ò « Ó× « Ý ¼ Ý ¼ Ó× « Ü ¼ ×Ò « (4.2.11) ¼ ¼ ½ ¿ 4.2.3 FORMULAE FOR SYMMETRIES: POLAR COORDINATES 1. Rotation around the origin through an angle «: ´Öµ ´Ö · «µ (4.2.12) 2. Re ec tion in a line through the origin and making an angle « with the positive Ü-axis: ´Öµ ´Ö ¾« µ (4.2.13) 4.2.4 CRYSTALLOGRAPHIC GROUPS A group of symmetries of the plane that is doubly in nite is a wallpaper group, or crystallographic group. There are 17 types of such groups, corresponding to 17 essentially distinct ways to tile the plane in a doubly periodic pattern. (There are also 230 three-dimensional crystallographic groups.) The simplest crystallographic group involves translations only (page 309, top left). The others involve, in addition to translations, one or more of the other types of symmetries (rotations, re ection s, glide-re ections). The Conway notation for crystallographic groups is based on the types of non-translational symmetries occurring in the “simplest description” of the group: 1. Æ indicates a translations only, 2. £ indicates a re ection (mirror symmetry), 3. ¢ a glide-re ection, 4. a number Ò indicates a rotational symmetry of order Ò (rotation by ¿¼ Æ Ò). In addition, if a number Ò comes after the £ , the center of the corresponding rotation lies on mirror lines, so that the symmetry there is actually dihedral of order ¾Ò. Thus the group ££ in the table below (page 309, middle left) has two inequivalent lines of mirror symmetry; the group ¿¿¿ (page 311, top right) has three inequivalent centers of order-3 rotation; the group ¾¾ £ (page 309, bottom right) has two inequivalent centers of order-2 rotation as well as mirror lines; and £ ¿¾ (page 311, bottom right) has points of dihedral symmetry of order ½¾´ ¾ ¢ µ, 6, and 4. The following table gives the groups in the Conway notation and in the notation traditional in crystallography. It also gives the quotient space of the plane by the © 2003 by CRC Press LLC
Page 1 and 2: Chapter Geometry 4.1 COORDINATE SY
Page 3 and 4: 4.19.2 Oblique spherical triangles
Page 5 and 6: FIGURE 4.1 Change of coordinates by
Page 7 and 8: 4.1.4.1 Relations between Cartesian
Page 9: This formula also covers passing fr
Page 13 and 14: Æ p1 ¢¢ pg ££ pm ¢£ cm ¾¾
Page 15 and 16: Æ p1 ¿¿¿ p3 £ ¿¿¿ p3m1 ¿
Page 17 and 18: (also known as a homothety) with an
Page 19 and 20: (In an oblique coordinate system, e
Page 21 and 22: 4.4.4 CONCURRENCE AND COLLINEARITY
Page 23 and 24: FIGURE 4.10 Notations for an arbitr
Page 25 and 26: FIGURE 4.12 Left: Ceva’s theorem.
Page 27 and 28: 2. The rhombus or diamond Ð , wher
Page 29 and 30: 4.6 CONICS A conic (or conic sectio
Page 31 and 32: FIGURE 4.17 Hyperbola with transver
Page 33 and 34: (c) If ¼, the equation again repr
Page 35 and 36: (a) The area of the ellipse is ¾
Page 37 and 38: 8. Let Ä be any line in the plane.
Page 39 and 40: FIGURE 4.18 The arc of a circle sub
Page 41 and 42: FIGURE 4.21 The semi-cubic parabola
Page 43 and 44: FIGURE 4.24 De ning property of the
Page 45 and 46: FIGURE 4.27 Left: initial con gurat
Page 47 and 48: FIGURE 4.29 The Bernoulli or logari
Page 49 and 50: 4.7.7 CLASSICAL CONSTRUCTIONS The a
Page 51 and 52: and the positive polar axis, and th
Page 53 and 54: mations of the following types: 1.
Page 55 and 56: 7. Re ec tion in a plane with equat
Page 57 and 58: 4.10.3 PROJECTIVE TRANSFORMATIONS A
Page 59 and 60: 6. The angle between two planes ¼
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Angle between lines with direction
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FIGURE 4.35 The Platonic solids. To
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FIGURE 4.36 Left: an oblique cylind
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For a frustum of a right circular c
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Given a general quadratic equation
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FIGURE 4.40 Left: a spherical cap.
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FIGURE 4.41 Right spherical triangl
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ØÒ ØÒ ØÒ ¾ ¾ ¾
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The angle between these vectors, «
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esults hold at point f´Øµ of :
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Asymptotic direction Asymptotic lin
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8. The principal directions Ù Ú
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4.21 ANGLE CONVERSION Degrees Radia
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Chapter 4: Geometry

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