1 Geometric Coordinate Systems

7/23/2019 1 Geometric Coordinate Systems

1/20

G43 Geometric Modeling

Geometric coordinate

1

Dr.C. ParamasivamE-mail: [email protected]


2/20

1. Define the coordinate system for the development

of models based on input and geometry.2. Develop and manipulate the curves and surfaces

using parametric equations.

.

modeling techniques.4. Implement the transformation and projection over

the geometric models.5.

Implement the neutral file formats over 2Dwireframe models.


3/20

1. Cartesian CoordinatesThe location of a point in three dimensional space may bespecified by an ordered set of numbers (x, y, z).


4/20

2. Cylindrical Coordinate

r


5/20

Cylindrical Coordinates

Cylindrical coordinates just adds a z-coordinateto the polar coordinates (r,).

r


6/20



r


7/20



r


8/20



r


9/20



r


10/20



r


11/20


r

(r,,z)

r


12/20

Convertion between rectangularand Cylindrical Coordinates

r

(r,,z) cos( )sin( )

x r

y r

z z

=

=

=

Cylindrical to rectangular

r

Rectangular to Cylindrical

2 2 2

tan( )

r x y

y

x

z z

= +

=

=


13/20

Cylindrical coordinates are a generalization of two-dimensionalpolar coordinates to three dimensions by a height along (z) axis.

A point P is represented by an

ordered triple of .( )zr ,,

To change from rectangular to cylindrical:


zz

x

y

yxr

=

=

+=

1

22

tan

zzry

rx

==

=

,sin

cos

To change fromcylindrical torectangular:


14/20

3. Spherical Coordinates

It is a coordinate system for three-dimensional space where theposition of a point is specified by three numbers: the radial distanceof that point from a fixed origin, its inclination angle measured from afixed zenith direction, and the azimuth angle of its orthogonal

projection.


15/20

In Cartesian CS, directions of unit vectors are independent of

their positions.

In Cylindrical and Spherical systems, directions of unit vectorsdepend on positions....

Cartesian Cylindrical Spherical


16/20

Spherical coordinates (r, , ) of a point can be obtainedfrom its Cartesian coordinates (x, y, z) by the formulas:

Cartesian coordinates may be retrieved from the spherical

coordinates (r, , ), where r

[0, ),

[0, ],

[0, 2 ), by:


17/20

Cylindrical coordinates (r, , z) may be converted into

spherical coordinates (, , ), by the formulas

S herical coordinates ma be converted into

cylindrical coordinates by the formulae


18/20

Applications:

The most common use of cylindrical coordinates is to give theequation of a surface of revolution. The z-axis is taken asthe axis of revolution.

Example:

Paraboloid of revolution might have equation z = r2.

Cartesian coordinate equation of the paraboloid of revolutionwould be z = x2 + y2.


19/20

x

y

yxr

1

22

tan

=

+=

Rectangular to cylindrical:

sin

cos

ry

rx

=

=Cylindrical torectangular:

Summary

Spherical to Rectangular: cos,sinsin,cossin === zyx

Rectangular to Spherical:

++

==++=222

222 arccos,tan,zyx

z

x

yzyx

Spherical to cylindrical ( ): cos,,sin222 === zr0r

Cylindrical to spherical ( ):0r

+

==+=22

22 arccos,,zr

zzr


20/20

1.What is Cartesian coordinate?

2.What is Polar coordinate?

3.What is cylindrical coordinate?

4.What is spherical coordinate?

5.How can you convert rectangular space point in to cylindrical

Sample Questions

space point?

6.How can you convert polar coordinate point in to cylindrical

space point?

7.Find the cylindrical coordinates of the point whose Cartesian

coordinates are (1, 2, 3)?

8.Find the Cartesian coordinates of the point whose cylindrical

coordinates are (2, Pi/4, 3)?

1 Geometric Coordinate Systems

Documents

Transcript of 1 Geometric Coordinate Systems