Page 484 - House of Maxell

464 APPENDIX

() = () + () + () (A.2)
0
0
0

a) b)
Figure A2 a) Trajectory of vector tip along curve, b) 2D parametric curve ()

= 1.5 cos() − cos(30), () = 1.5 sin() − sin(30)), 0 ≤ ≤ 4

Then each point along the curve is specified by three functions that describe the curve in terms
of the parameter t. The plots in Figure A2b are exemplary.

Parametric Curved Surface

One more time we come across the problem what is the less restrictive way to define a surface.
Theoretically, any coordinate system is acceptable, but as in the case of the curve, more freedom
gives the parametric representation of the surface. Definitely, we need now a pair of scalar
variables instead of one as in the event of the curve. Following this idea the position vector for
a curved surface can be written as a function of two independent variables u and

(, ) = (, ) + (, ) + (, ) (A.3)
0
0
0
Here each combination of u and corresponds some point on the surface as it is displayed on
the plot in Figure A3 where the surface is formed by joining surface patches. The red arrows
equation is (, ) = + + (6 + | sin( + ) + cos ( − )|).
0
0
0

Figure A3 Illustration of parametric representation of surface