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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.

y

x

Dimensions

a 0 b

y

x

y

x

z

1D

2D

3D

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.

Affine space

V – Vector space – nonempty set of points

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Affine space

V – Vector space – nonempty set of points

"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

There exists point , such that

:

is one-to-onecorrespondence

P P V

v B A

O

P V a A O

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

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.

;O V O w w W

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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

v = B

w2 = B – A

w = u + v

dotprod = u*v

mu = Length[u]

Magnitude of the cross product (with sgn) crossprod = u ⊗ w

Radius vector

Vectors in R3

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

Linear combination, linear dependence

Vector subspace

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Linear combination

Let V is a vector space over a field R.

Vector x V is given by ordered tuple. 1 2 3, , , , nx x x x x

1 1 2 2

1

n

i i n n

i

x a e a e a e a e

Vector x is a linear combination of set ( e1, … , en ), iff there exists

n-tuple ( α1, … , αn ) real numbers which yields

Affine combination

Take an arbitrary point A in affine space (O,V)

1

1 1 2 2

1 1 2 2

1

; where

... 1

n

i i i i

i

n n

n

n n i

i

A O x O a e e E O

A O a E O a E O a E O

A a E a E a E O a

Affine combination of points O, E1,…, En

1 1 2 2

1

1 1 2 2 0

0

... 1

... , where 1

n

n n i

i

n

n n i

i

A a E a E a E O a

A a E a E a E a O a

Arbitrary point A in affine space (O,V) could be expressed

as an affine combination

Convex combination of points O, E1,…, En

- linear combination of points where all coefficients are non-

negative and sum to 1

1 1 2 2 0

0

... , where 0 and 1n

n n i i

i

A a E a E a E a O a a

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Convex combination of points O, E1,…, En

1

4

1 1 2 2 3 3 4 4 0

0

, where 0i i

i

A a E a E a E a E a O a a

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

0

y mx q

X A t u

ax by cz d

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Parametric equations of a line

X1

A

u X1 = A + u X2

X2 = A + 2.u

X3

X3 = A + 3.u X4

X4 = A + 1/2 . u

X5

X5 = A + (-1) . u X6 X6 = A + (- 3/2) . u

• 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

RtutAXpX ;

p

Příklad

GeoGebra-primka.ggb

Task Determine the parametric form for the line AB.

u = B – A

C = A+t*u

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The slope of a line m = rise over run.

Linear function y = mx + q

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.

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v

X1

X2

X3 = A + 2u + 3v

u 2u

X1 = A + 2u

X2 = A + 3v

X3

3v 2u + 3v

A

Parametric form for the plane in 3D space

u = B – A

v = C – A

X = A+t*u+s*v

C

B

• 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

A

v

u

RstvsutAXX ,;

Vektorový součin

Parametric form for the plane in 3D space

X3

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Perpendicular (normal) vector of p

p

];[ 21 ppP

];[ yxX

X Arbitrary point on the line p

);( ban

n

)( PX

nPXpX

(for X≠P)

0 nPX

021 bpyapx

021 bpapbyax cbpap 21

0 cbyaxpX

label:

General equation of the hyperplane in 2D space

],,[ 321 pppP

],,[ zyxX

),,( cban

)( PX

(for X≠P)

0 nPX

0321 cpbpapczbyax

dcpbpap 321

0 dczbyaxX

0321 cpzbpyapx

label:

nPXX

General equation of the hyperplane in 3D space

Perpendicular (normal) vector of p

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Conic Sections

Where do you see conics in real life?

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• A circle with center (h, k) and radius r has length

to some point (x, y) on

the circle.

• Squaring both sides yields the center-radius

form of the equation of a circle.

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.

22 )()( kyhxr

22)()( kyhxr

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.

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

. 2 2( ) ( )

2x h y k r

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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.

22

222

)4()3(36

)4())3((6

yx

yx

General Form of the

Equation of a Circle

For real numbers c, d, and e, the equation

can have a graph that is a circle, a point, or is empty.

22x y cx dy e 0

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Parametric equations for the circle

2 2 1x y

cos

sin ; 0,2

x r t

y r t t

2 2 2

2 2 2 2 2cos sin

x y r

r t r t r

Parametric equations for the circle

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Parabola

http://tube.geogebra.org/

Equations and Graphs of Parabolas

• For example, let the directrix be the line y = –c and

the focus be the point F with coordinates (0, c).

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.

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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.

cyx

cycycycyx

cycycycyx

cyxxcyx

DPdFPd

4

22

22

))(()()()0(

),(),(

2

22222

22222

2222

Parabola with a Vertical Axis and Vertex (0, 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.

The endpoints of the chord are ( x, c), so the length is |4c|.

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.

c cxcx

cyx

2or 24

4

22

2

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Determining Information about Parabolas

from Equations

Example Find the focus, directrix, vertex, and axis

of each parabola.

(a)

Solution

(a)

xyyx 28 (b)8 22

2

84

c

c

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.

(b)

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.

7

284

c

c

Determining Information about

Parabolas from Equations

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An Application of Parabolas

Example The Parkes radio tele-

scope 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 ).32,105(32,2210

2

2

2

32 (105)32 32

The cross section can be described by105 11,025

32.

11,025

a

a

y x

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(b) Since

The receiver should be placed at (0, 86.1), or

86.1 feet above the vertex.

,025,11

32 2xy

.1.86128

025,11

32

025,114

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c

c

ac

An Application of Parabolas

Trajectory of a projectile

path that a thrown or

launched projectile or

missile

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Ellipse

GeoGebra-kuzelosecky.ggb

x = a · cos t + m

y = b · sin t + n

where t is a polar angle

between radius vector of X

and x axis

t<0;2π).

Parametric equations of the ellipse

X[x;y]

S[m;n]

y

x 0

m x

n

y

× t

2 2

2 2

2 2

2 2

2 2 2 2

2 2

1

.cos .sin1

. cos . sin1

x m y n

a b

a t m m b t n n

a b

a t b t

a b