16.360 Lecture 15

• Today

Transformations between coordinate systems

1. Cartesian to cylindrical transformations2. Cartesian to spherical transformations3. Cylindrical to spherical transformations4. Distance between two points

16.360 Lecture 15

• Cartesian to cylindrical transformation

,22 yxr

),(tan 1

x

y

),cos(rx

),sin(ry

16.360 Lecture 15

• Cartesian to cylindrical transformation),cos(ˆˆ xr ),sin(ˆˆ yr

),sin(ˆˆ x ),cos(ˆˆ y

,cosˆsinˆˆ yx

),cos(ˆ)ˆˆ(ˆˆ axybxaxr

),sin(ˆ)ˆˆ(ˆˆ byybxayr

,sinˆcosˆˆ yxr

,sinˆcosˆˆ rx

,cosˆsinˆˆ ry

16.360 Lecture 15

• Cartesian to cylindrical transformation

),cos(ˆˆ xr ),sin(ˆˆ yr

),sin(ˆˆ x ),cos(ˆˆ y

,sincos yxr AAA

,ˆˆˆ zAyAxAA zyx

,ˆˆˆ zAArAA zr

,cossin yx AAA

,sincos AAA rx

,cossin AAA ry

16.360 Lecture 15

• Cartesian to Spherical transformation

,222 zyxr

),(tan22

1

z

yx

),(tan 1

x

y

,cossin Rx

,sinsin Ry

,cosRz

16.360 Lecture 15

• Cartesian to Spherical transformation

,ˆˆˆ bzarR

,sinˆˆ bzR

,cosˆˆ arR ,sinˆcosˆˆ yxr

,cosˆsinsinˆcossinˆˆ zyxR

,cosˆˆ cr,ˆˆˆ dzcr ,sinˆˆ cr

,sinˆsincosˆcoscosˆˆ zyx

,cosˆsinˆˆ yx

16.360 Lecture 15

• Cartesian to Spherical transformation

,ˆˆˆˆ edcRx

,sinˆˆ bzR

,cossinˆˆ xR

,sinˆcosˆˆ yxr

,sinˆcoscosˆcossinˆˆ Rx

,cosˆˆ cr,ˆˆˆ dzcr ,sinˆˆ cr

,sinˆcosˆˆ Rz

,coscosˆˆ x ,sinˆˆ ex

,cosˆsincosˆsinsinˆˆ Ry

16.360 Lecture 15

• Distance between two points:

,)()()(2/12

122

122

121212 zzyyxxRd

,)()sinsin()coscos(2/12

122

11222

11221212 zzrrrrRd

,)cos(sinsincos[cos2

,)coscos()sinsinsinsin()cossincossin(2/1

121212212

122

2/121122

2111222

21112221212

RRRR

RRRRRRRd