Rigid Motions (Isometries)
Section 2.1 discusses the vectors and their use in analytic geometry. In Section 2.2 we study rigid motions of \({\mathbb{R}}^{n}\) (n≥2), including the complex and quaternionic approaches. Section 2.3 is devoted to the geometry on a sphere (induced by Eu
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Rigid Motions (Isometries)
Section 2.1 discusses vectors and their use in analytic geometry. In Section 2.2 we study rigid motions of Rn (n ≥ 2), including the complex and quaternionic approaches. Section 2.3 is devoted to geometry on a sphere (induced by Euclidean geometry of space), and also introduces stereographic projection. The study of polyhedra is organized in Section 2.4 into a sequence of themes: the notion of a polyhedron, Platonic solids, symmetries of geometrical figures, star-shaped polyhedra, and Archimedean solids. Section 2.5 (Appendix) surveys matrices and groups.
2.1 Vectors A vector in Rn (as a point of Rn ) is an ordered set x = (x1 , x2 , . . . , xn ) of n real numbers. The origin O in Rn is the zero vector.
2.1.1 Vectors in R3 If a1 , . . . , an are vectors, and if λ1 , . . . , λn are real numbers, then the linear combination λ1 a1 + · · · + λn an of vectors is again a vector. The directed line segment from the point a to the point b is the set of points [a, b] = {a + t(b − a) : 0 ≤ t ≤ 1} with initial point a and final point b. A nonempty set X ⊂ R3 is called convex if X contains, with any two points x, y, also the line segment [x, y]. R V. Rovenski, Modeling of Curves and Surfaces with MATLAB , Springer Undergraduate Texts in Mathematics and Technology 7, DOI 10.1007/978-0-387-71278-9 2, c Springer Science+Business Media, LLC 2010
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2 Rigid Motions (Isometries)
Triangles, rectangles, and disks in a plane are examples of convex sets. Recall that for all x ∈ R3 , x = x1 i + x2 j + x3 k, where i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) are the unit vectors; these points lie at a unit distance along the three coordinate axes. The scalar product of the vectors x and y is x · y := x1 y1 + x2 y2 + x3 y3 . We call x = x21 + x22 + x23 the norm (length) of x. The following properties of the scalar product are immediate: (1) (2) (3) (4)
(λ1 x + λ2 y) · z = λ1 (x · z) + λ2(y · z); x · y = y · x; x − y2 = x2 + y2 − 2(x · y); i · j = j · k = k · i = 0, and i · i = j · j = k · k = 1.
We calculate the angle θ between the vectors x, y by cos θ = x · y/(x · y). The triangle inequality. x − z ≤ x − y + y − z, x, y, z ∈ R3 . The vector product of the two vectors x and y is x × y := (x2 y3 − x3 y2 ) i + (x3y1 − x1 y3 ) j + (x1y2 − x2 y1 ) k. The following properties of the vector product are immediate: (1) (2) (3) (4)
x × y is orthogonal to x and to y; (λ1 x + λ2 y) × z = λ1 (x × z) + λ2(y × z); x × y = −y × x; x × y = 0 if and only if x, y are collinear. In particular, x × x = 0.
The scalar triple product of the vectors x, y, and z& is (x, y,&z) := x · (y × z). Hence, &x1 x2 x3 & & & the scalar triple product is the determinant (x, y, z) = &&y1 y2 y3 &&. The vector triple prod& z1 z2 z3 & uct of the vectors x, y, and z is [x, y, z] := x × (y × z). Consider two nonzero vectors a and b, in the plane x3 = 0. Let θ ∈ (0, π ) be the angle between a and b, measured in the counterclockwise direction. We can easily see that Δ (a, b) := u1 v2 − u2 v1 = a · b sin θ . Following
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