Some Material on Swept Solids and Surfaces of Revolution

From Chapter 10 of Mortenson

Sections 10.5 – 10.6

Geometric Modeling91.580.201

Scaling Transformations

affect geometry but not topology of object

primitive shapes

Source: Mortenson, Ch 10

Differential Scaling Transformations (continued)

restrictions

Source: Mortenson, Ch 10

Differential Scaling Transformations (continued)

Sample restrictions: thtaabtlhba 4,2,,0,,,,

Source: Mortenson, Ch 10

Parameterized Shape of Variable Topology

Source: Mortenson, Ch 10

Sweep Solids• Moving an object along a path.

– Generator = sweeping object: curve, surface, or solid– Director = path

• Common for modeling constant cross-section mechanical parts.

• Translational sweep (extrusion): moving a planar curve or planar shape along a straight line normal to plane of curve.

– More generally, sweep one curve along another.

• Rotational sweep: rotating a planar curve or shape (with finite area) about an axis.

Source: Mortenson, Ch 10

Sweep Solids (continued)

Source: Mortenson, Ch 10

some problematic situations

Loss and Eshleman (1974) Position and Direction Specification for Swept Solids

Source: Mortenson, Ch 10

trimming

Loss and Eshleman (1974) Position and Direction Specification for Swept Solids

(continued)

Source: Mortenson, Ch 10

Surfaces of Revolution

)(sin)(cos)(),( uuuu zxxp

Source: Mortenson, Ch 10

)()()( uuu zxp

Example: z-axis of rotation

Surfaces of Revolution (continued)

Source: Mortenson, Ch 10

More general example using cubic Hermite curve: goal is to find a Hermite patch describing the surface.

axis of rotation

Hermite curve

angle of rotation