16.360 lecture 15 today transformations between coordinate systems 1.cartesian to cylindrical...
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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),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