Angle and Slope

and Isometri

Angle and Slope

Concept of distance point (x1, y1) and (x2, y2) are written in

algebraic function :

yyxx 1212

22

y2 – y1

x2 – x1

y2

y1

x1 x2

K L

M

yyxx 1212

22

So, Length of KM

y

x0

tt

tt

21

21

1

21

21

21tantan1

tantantan

Equality isn’t case for concept of angle so to stay within ofalgebra, we need work with the slope rather than angle Ɵ.

Lines make same angle with x-axis if they have same slope.

To test equality of angles in general we need the concept of

relative slope : If line l1 has slope t1 and line l2 has slope t2 then

slope of l1 relative to l2 is defined to be:

Isometri

A transformation of plane is a function that sends points to

points, it written in simply a function f: R2 →R2

A transformation f is called an isometry.

If points p1 and p2 sends to points f(p1) and f(p2) for the

same distance apart.

Therefore

|f(p1)f(p2)|= |p1p2| is an isometry function f with property

for any two points p1, p2.

Translate from x to yX y

A translations moves each point of the plane the

same distance in the same direction.

Translations

Translations

Each translation depends on two constants a, b is denoted by ta,b.

So each point (x,y) moves to (x+a, y+b)

To proof it, let p1 = (x1, y1) and p2 = (x2, y2)

ta,b (p1) = (x1+a, y1+b) and ta,b p2 = (x2+a, y2+b)

bybyaxaxptpt baba 1212

22

2,1,

yyxx 1212

22

pp21

1+2

2+3

5

2

1 3

Example:

Construct t2,1 (A,B,C) !

A(1,2)

C(3,5)

y

x0

B(3,2)

t2,1 (A) = (x+2, y+1)

t2,1 (A) = (1+2, 2+1)

= (3, 3)

t2,1(B) = (x+2, y+1)

t2,1 (B) = (3+2, 2+1)

= (5, 3)

t2,1 (C) = (x+2, y+1)

t2,1 (C) = (3+2, 5+1)

= (5, 6)

A’(3,3)

C’(5,6)

B’(5,3)

Rotation

A rotation is a transformation that turning a

figure around a point or a line.

Rotations

The rotations rc,s sends point (x,y) to the point (cx-sy, sx+cy)

let p1 = (x1, y1) and p2 = (x2, y2)

rc,s (p1) = (cx1- sy1, sx1+cy1) rc,s (p2) = (cx2-sy2, sx2+cy2)

therefore

12121212

22

2,1,yycxxsyysxxcprpr scsc

1212

222

222

yyxx scsc

1212

22

yyxx

pp21

2.1+3.2

7

2

1

5

Example:

Construct r2,1 (A, B, C) !

r2,1 (A) = (2x-1y, 1x+2y)

r2,1 (A) = (2.1-1.2, 1.1+2.2)

= (0, 5)

B(3,2)

y

rc,s (p1) = (cx- sy, sx+cy)

x0

r2,1 (C)= (2x-1y, 1x+2y)

r2,1 (C) = (2.3-1.4, 1.3+2.4)

= (2, 11)

r2,1 (B) = (2x-1y, 1x+2y)

r2,1 (B) = (2.3-1.2, 1.3+2.2)

= (4, 7)

A(1,2)

C(3,4)

3 4

B’(4,7)

A’(0,5)

C’(2,11)11

2

Reflections

Reflection sends P = (x,y) to = (x, -y)p

Here combining reflection in x-axis with translation and rotation

For example :

Reflection in the line y = 1 is result of three isometries:

• t0,-1 a translation that moves the line y = 1 to the x-axis

• reflection in the x-axis

• t0,1 which moves the x-axis back to the line y = 1

t2,31

-1

Example:

Reflect Q (2,3) in the x-axis !

Q(5,1)

y

x0

)1,5(Q

Reflection sends Q = (x,y) to = (x, -y)

In the x-axix

Q

Q = (5,1) → )1,5(Q

Glide Reflections

A glide reflection is the result of a reflection followed by a

translation in the direction of the line of reflection.

For example:

If we reflect in the x-axis, sending (x,y) to (x, -y) and follows

with t1,0 then it sends to (x=1, -y)

A glide reflection with nonzero translation length is different

from the three types of isometry prevously considered:

• It’s not a translation because a translation maps any line in the direction of

translation into itself, whereas a glide reflection maps only one line into itself

•It’s not a rotation because a fixed point, whereas a glide reflection doesn’t.

•It’s not a reflection because it has fixed point

Thank You