y x Analytic Geometry Analytic geometry, usually called coordinate geometry or analytical geometry, is the study of geometry using the principles of algebra The link between algebra and geometry was made possible by the development of a coordinate system which allowed geometric ideas, such as point and line, to be described in algebraic terms like real numbers and equations. Central idea of analytic geometry relate geometric points to real numbers. Affine space V Vector space nonempty set of points Dimensions a 0 b y D 1D x By defining each point with a unique set of real numbers, geometric figures such as lines, circles, and conics can be described with algebraic equations. z 3D y x 1
Affine space "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps Marcel Berger V Vector space nonempty set of points P P V v B A There exists point O, such that P V : a A O is one-to-onecorrespondence Affine subspace Let us consider an affine space A and its associated vector space V. Affine subspaces of A are the subsets of A of the form O V O w; ww where O is a point of A, and V a linear subspace of W. The linear subspace associated with an affine subspace is often called its direction, and two subspaces that share the same direction are said parallel.
Vectors in R Magnitude of the vector is equal to the distance of head and tail points. A = (1,3) B = (3,1) u = A Radius vector v = B w = B A w = u + v dotprod = u*v m u = Length[u] Magnitude of the cross product (with sgn) crossprod = u w Vectors in R 3 Linear combination, linear dependence Vector subspace A = (1,3,) B = (3,1,0) O = (0,0,0) u = A v = B w = u + v a = Plane[A,B,O] b = PerpendicularLine[O,a] C = Point[b] n = C dotprod = n*u 3
Linear combination Let V is a vector space over a field R. Vector x V is given by ordered tuple. n x a e a e a e a e i1 i i 1 1 n n x x1, x, x3,, xn Vector x is a linear combination of set ( e 1,, e n ), iff there exists n-tuple ( α 1,, α n ) real numbers which yields Affine combination Take an arbitrary point A in affine space (O,V) n A O x O a e ; where e E O i1 1 1 i i i i A O a E O a E O a E O n A a1 E1 ae... anen O1 ai i1 n n Affine combination of points O, E 1,, E n Arbitrary point A in affine space (O,V) could be expressed as an affine combination n A a1e1 ae... anen O1 ai i1 A a E a E... a E a O, where a 1 1 1 n n 0 i i0 Convex combination of points O, E 1,, E n - linear combination of points where all coefficients are nonnegative and sum to 1 n n A a E a E... a E a O, where a 0 and a 1 1 1 n n 0 i i i0 4
Convex combination of points O, E 1,, E n 4 A a1 E1 ae a3e3 a4e4 a0oai, where ai 0 i0 1 Straight line in two-dimensional space A straight line is unambiquously determined by two different points. A straight line can be analytically expressed in Slope form y mx q Parametric form X A t u General equation ax by cz d 0 5
Parametric equations of a line All points X = A + t.u where t R form a line and vice versa -- all points on that line have the form X = A + t.u for some real number t. u direction vector X 3 p u X X 1 = A + u X = A +.u X 5 X 6 A X 1 X 4 X p X A t u ; t R X 3 = A + 3.u X 4 = A + 1/. u X 5 = A + (-1). u X 6 = A + (- 3/). u Task Příklad Determine the parametric form for the line AB. u = B A C = A+t*u GeoGebra-primka.ggb 6
Linear function y = mx + q The slope of a line m = rise over run. Calculating Slope Slope (m) = rise (change in y) / run (change in x) Rise is the vertical change and run is the horizontal change Run (3) Rise (3) M = y/x M = 3/3 M = 1 The slope is 1. This means that for every increase of 1 on the x axis, there will be an increase of 1 on the y axis. 7
Parametric form for the plane in 3D space X X 3 C 3v u + 3v v A u B u X 1 u = B A v = C A X = A+t*u+s*v X 1 = A + u X = A + 3v X 3 = A + u + 3v Parametric form for the plane in 3D space All points X = A + t.u + s v, where t, s R form a plane and vice versa -- all points on that plane have the form X = A + t.u + s v for some real numbers t,s. u, v direction vectors v X 3 A u X X A t u sv; t, s R Vektorový součin 8
General equation of the hyperplane in D space X p ax by c 0 p n ( a; b) ( X P) X[ x; y] X p P[ p 1 ; p] X n X P n (for X P) X Pn 0 x p1 a y p b 0 ax by ap bp 0 1 Arbitrary point on the line p Perpendicular (normal) vector of p label: ap bp c 1 General equation of the hyperplane in 3D space n ( a, b, c) X ax by cz d 0 ( X P) X[ x, y, z] P[ p1, p, p3] X X P n X Pn 0 x p a y p b z p (for X P) 1 3c 0 ax by cz ap1 bp cp3 0 label: ap bp cp d 1 3 Perpendicular (normal) vector of p 9
Conic Sections Where do you see conics in real life? 10
Circles A circle is a set of points in a plane that are equidistant from a fixed point. The distance is called the radius of the circle, and the fixed point is called the center. A circle with center (h, k) and radius r has length r ( x h) ( y k) to some point (x, y) on the circle. Squaring both sides yields the center-radius form of the equation of a circle. r ( x h) ( y k ) Center-Radius Form of the Equation of a Circle The center-radius form of the equation of a circle with center (h, k) and radius r is ( x h) ( y k) r. Notice that a circle is the graph of a relation that is not a function, since it does not pass the vertical line test. 11
Finding the Equation of a Circle Example Find the center-radius form of the equation of a circle with radius 6 and center ( 3, 4). Graph the circle and give the domain and range of the relation. Solution Substitute h = 3, k = 4, and r = 6 into the equation of a circle. 6 ( x ( 3)) 36 ( x 3) ( y 4) ( y 4) General Form of the Equation of a Circle For real numbers c, d, and e, the equation x y cx dy e 0 can have a graph that is a circle, a point, or is empty. 1
Parametric equations for the circle x y 1 Parametric equations for the circle x r cos t y r sin t; t 0, x y r r cos t r sin t r 13
Parabola http://tube.geogebra.org/ Equations and Graphs of Parabolas A parabola is a set of points in a plane equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line the directrix, of the parabola. For example, let the directrix be the line y = c and the focus be the point F with coordinates (0, c). 14
Equations and Graphs of Parabolas To get the equation of the set of points that are the same distance from the line y = c and the point (0, c), choose a point P(x, y) on the parabola. The distance from the focus, F, to P, and the point on the directrix, D, to P, must have the same length. ( x 0) x y x y d( P, F) d( P, D) ( y c) ( x x) ( y ( c)) yc c y yc c yc c y yc c x 4cy Parabola with a Vertical Axis and Vertex (0, 0) The parabola with focus (0, c) and directrix y = c has equation x = 4cy. The parabola has vertex (0, 0), vertical axis x = 0, and opens upward if c > 0 or downward if c < 0. The focal chord through the focus and perpendicular to the axis of symmetry of a parabola has length 4c. Let y = c and solve for x. x 4cy x 4c x c or c The endpoints of the chord are ( x, c), so the length is 4c. 15
Determining Information about from Equations Parabolas Example Find the focus, directrix, vertex, and axis of each parabola. (a) x 8y (b) y 8x Solution (a) 4c 8 c Since the x-term is squared, the parabola is vertical, with focus at (0, c) = (0, ) and directrix y =. The vertex is (0, 0), and the axis is the y-axis. Determining Information about Parabolas from Equations (b) 4c 8 c 7 The parabola is horizontal, with focus ( 7, 0), directrix x = 7, vertex (0, 0), and x-axis as axis of the parabola. Since c is negative, the graph opens to the left. 16
An Application of Parabolas Example The Parkes radio telescope has a parabolic dish shape with diameter 10 feet and depth 3 feet. Because of this parabolic shape, distant rays hitting the dish are reflected directly toward the focus. An Application of Parabolas (a) (b) Determine an equation describing the cross section. The receiver must be placed at the focus of the parabola. How far from the vertex of the parabolic dish should the receiver be placed? Solution (a) The parabola will have the form y = ax (vertex at the 10,3 (105,3 origin) and pass through the point ). 3 a(105) 3 3 a 105 11,05 3 y x. 11,05 The cross section can be described by 17
An Application of Parabolas (b) Since 3 y x, 11,05 1 4c a 4c c 11,05 3 11,05 18 86.1. The receiver should be placed at (0, 86.1), or 86.1 feet above the vertex. Trajectory of a projectile path that a thrown or launched projectile or missile 18
Ellipse GeoGebra-kuzelosecky.ggb Parametric equations of the ellipse y y n 0 t S[m;n] m x m y n a x X[x;y] a.cos t m m b.sin t n n x a. cos t b. sin t a a b b 1 1 b x = a cos t + m y = b sin t + n where t is a polar angle between radius vector of X and x axis 1 t<0;π). 19