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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.
Dimensions
Affine space
V – Vector space
 – nonempty set of points
a
0
b
1D
y
2D
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-one correspondence
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; w  W 
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.
2
Vectors in R2
• 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
w2 = B – A
w = u + v
dotprod = u*v
mu = Length[u]
Magnitude of the cross product (with sgn)
crossprod = u ⊗ w
Vectors in R3
Linear combination, linear dependence
Vector subspace
A = (1,3,2)
B = (3,1,0)
O = (0,0,0)
u = A
v = B
w = u + 2v
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.
x   x1 , x2 , x3 ,
, xn 
Vector x is a linear combination of set ( e1, … , en ), iff there exists
n-tuple ( α1, … , αn ) real numbers which yields
n
x   ai ei  a1e1  a2 e2 
 an en
i 1
Affine combination
Take an arbitrary point A in affine space (O,V)
n
A  O  x  O   ai ei ; where ei  Ei  O
i 1
A  O  a1  E1  O   a2  E2  O  
 an  En  O 
n


A  a1 E1  a2 E2  ...  an En  O 1   ai 
i 1


Affine combination of points O, E1,…, En
Arbitrary point A in affine space (O,V) could be expressed
as an affine combination
n


A  a1 E1  a2 E2  ...  an En  O 1   ai 
i 1


n
A  a1 E1  a2 E2  ...  an En  a0 O, where  ai  1
i 0
Convex combination of points O, E1,…, En
- linear combination of points where all coefficients are nonnegative and sum to 1
n
A  a1 E1  a2 E2  ...  an En  a0 O, where ai  0 and  ai  1
i 0
4
Convex combination of points O, E1,…, En
1
 4 
A   a1 E1  a2 E2  a3 E3  a4 E4  a0 O    ai  , where ai  0
 i 0 
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
– Parametric form
– General equation
y  mx  q
X  A  t u
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
p
X3
u
X2
X1 = A + u
X2 = A + 2.u
X1
X4
X3 = A + 3.u
X4 = A + 1/2 . u
A
X5 = A + (-1) . u
X6
X6 = A + (- 3/2) . u
X5

X  p  X  A t u ; t R
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
M = y/x
M = 3/3
Rise
(3)
Run
(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
X2
3v
X3
2u + 3v
C
v
u
A
B
2u
X1
X1 = A + 2u
u = B – A
v = C – A
X = A+t*u+s*v
X2 = A + 3v
X3 = A + 2u + 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
X3

v
A
u


X    X  A  t  u  s  v; t , s  R
Vektorový součin
8
General equation of the hyperplane in 2D space
X  p  ax  by  c  0
X [ x; y]
( X  P)

n (a; b)
X

n
P[ p1; p2 ]

X  p   X  P  n
 X  P n  0
p
Arbitrary point on the line p
Perpendicular (normal) vector of p
(for X≠P)
x  p1  a   y  p2  b  0
ax  by   ap1  bp2   0
label:
 ap1  bp2   c
General equation of the hyperplane in 3D space

n (a, b, c)
X    ax  by  cz  d  0

X [ x, y, z ]
( X  P)
P[ p1 , p2 , p3 ]

X     X  P  n
 X  P n  0
Perpendicular (normal) vector of p
(for X≠P)
x  p1  a   y  p2  b  z  p3  c  0
ax  by  cz   ap1  bp2  cp3   0
label:
 ap1  bp2  cp3   d
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)2  ( y  k )2 to some point (x, y) on
the circle.
Squaring both sides yields the center-radius
form of the equation of a circle.
r 2  ( x  h)2  ( y  k )2
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)2  ( y  k )2  r 2 .
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.
62  ( x  (3)) 2  ( y  4) 2
36  ( x  3) 2  ( y  4) 2
General Form of the
Equation of a Circle
For real numbers c, d, and e, the equation
x 2  y 2  cx  dy  e  0
can have a graph that is a circle, a point, or is empty.
12
Parametric equations for the circle
x2  y 2  1
Parametric equations for the circle
x  r cos t
y  r sin t ; t   0, 2 
x2  y 2  r 2
r 2 cos 2 t  r 2 sin 2 t  r 2
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.
d ( P, F )  d ( P, D )
( x  0)  ( y  c) 2  ( x  x) 2  ( y  (c)) 2
x 2  y 2  2 yc  c 2  y 2  2 yc  c 2
x 2  y 2  2 yc  c 2  y 2  2 yc  c 2
x 2  4cy
2
Parabola with a Vertical Axis and Vertex (0, 0)
The parabola with focus (0, c) and directrix y = –c has
equation x2 = 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 2  4cy
x 2  4c 2

x  2c or 2c
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.
(b) y 2  28 x
(a) x 2  8 y
Solution
(a) 4c  8
c2
Since the x-term is squared, the
parabola is vertical, with focus
at (0, c) = (0, 2) and directrix
y = –2. The vertex is (0, 0), and
the axis is the y-axis.
Determining Information about
Parabolas from Equations
(b)
4c  28
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 210 feet and depth
32 feet. Because of this parabolic
shape, distant rays hitting the dish
are reflected directly toward the focus.
An Application of Parabolas
(a) Determine an equation describing the cross section.
(b) 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 = ax2 (vertex at the
origin) and pass through the point  210
2 , 32   (105, 32).
32  a(105) 2
32
32
a

The cross section can be described by
2
105 11,025
32
y
x2.
11,025
17
An Application of Parabolas
(b) Since y 
32 2
x ,
11,025
1
a
11,025
4c 
32
11,025
c
 86.1.
128
4c 
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
x = a · cos t + m
X[x;y]
y
y = b · sin t + n
t
×
S[m;n]
n
m
x
x
0
where t is a polar angle
between radius vector of X
and x axis
 x  m
a
2
2

 y  n
b
2
 a.cos t  m  m 
2
2
t<0;2π).
1
 b.sin t  n  n 

a2
a.2 cos 2 t b.2 sin 2 t

1
a2
b2
b2
2
1
19