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**Last updated: Yeh-Liang Hsu (2010-10-29).**

**Note: This is the course material for “ME550 Geometric modeling and
computer graphics,” Yuan Ze University. Part of this material is adapted from ***CAD/CAM
Theory and Practice*, by Ibrahim Zeid, McGraw-Hill, 1991. This material is
be used strictly for teaching and learning of this course.

# Analytic surfaces

## 1.
Surface models

### Plane surface

### Ruled (lofted) surface

### Surface of revolution

### Tabulated cylinder

**◇****Assignment 1**

### Bezier surface

### B-spline surface

### Coons patch

**◇****Assignment 2**

### Fillet surface

### Offset surface

**◇****Assignment 3**

## 2.
Parametric representations of
analytic surfaces

### Plane surface

### Ruled surface

**◇****Assignment 4**

**◇****Assignment 5**

### Surface of revolution

**◇****Assignment 6**

### Tabulated Cylinder

Shape design and
representation of complex objects such as car, ship and airplane bodies as well
as castings cannot be achieved utilizing wireframe modeling. In such cases, **surface modeling must be utilized to
describe objects precisely and accurately**. Due to
the richness in information of surface models, their use in engineering and
design environments can be extended beyond just geometric design and
representation. **They are
usually used in various applications such as calculating mass properties,
checking for interference between mating parts, generating cross-sectioned
views, generating finite element meshes, and generating NC tool paths for
continuous path machining.**

A surface model
of an object is a more complete and less ambiguous representation that its
wireframe model. **Surface
models** take the modeling of an object one step
beyond wireframe models by **providing information on surfaces connecting the object edges**. **Visualization of a surface is aided by the addition of
artificial fairing lines (called mesh)**, which
criss-cross the surface and so break it up into a network of interconnected
patches. Surface models provide **hidden line** and surface algorithms
to add realism to the displayed geometry. **Shading algorithms** are only
available for surface and solid models.

There are two
types of surfaces. **Analytic surfaces are based on wireframe entities, and include
the plane surface, ruled surface, surface of revolution, and tabulated
cylinder. Synthetic surfaces are formed from a given set of data points or
curves and include the bicubic, Bezier, B-spline, and Coons patches**.

This is the simplest surface. It requires three non-coincident points to define an infinite plane. The plane surface can be used to generate cross-sectional views by intersecting a surface model with it, generate cross sections for mass property calculations, or other similar applications where a plane is needed. Figure 1 shows a plane surface.

Figure 1. Plane surface

This is a linear surface. As shown in Figure 2, a ruled surface interpolates linearly between two boundary curves that define the surface (rails). Rails can be any wireframe entity.

Figure 2. Ruled surface

This is an axisymmetric surface that can model axisymmetric objects. It is generated by rotating a planar wireframe entity in space about the axis of symmetry a certain angle (Figure 3).

Figure 3. Surface of revolution

This is a
surface generated by translating a planar curve a certain distance along a
specified direction (axis of the cylinder) as shown in Figure 4.

Figure 4. Tabulated cylinder

Construct the analytical surfaces discussed above using
your CAD software, if applicable. Describe the procedure for constructing these
surfaces. ◇

This is a surface that approximates given input data (Figure 5). It is different from the previous surfaces in that it is a synthetic surface, it does not pass through all given data points.

Figure 5. Bezier surface

This is a surface that can approximate or interpolate given input data (Figure 6). It is a synthetic surface. It is a general surface like the Bezier surface but with the advantage of permitting local control of the surface.

Figure 6. B-spline surface.

The above
surfaces are used with either open boundaries or given data points. The **Coons patch is used to create a
surface using curves that form closed boundaries** (Figure 7).

Figure 7. Coons patch

Construct the synthetic surfaces discussed above using
your CAD software, if applicable. Describe the procedure for constructing these
surfaces. ◇

This is a B-spline surface that blends two surfaces together (Figure 8).

Figure 8. Fillet surface.

Existing surfaces can be offset to create new ones identical in shape but may have different dimensions. For example, as shown in Figure 9, to create a hollow cylinder, the outer or inner cylinder can be created using a cylinder command and the other one can be created by an offset command.

Figure 9. Offset surface

Construct the fillet and offset surfaces discussed above
using your CAD software, if applicable. Describe the procedure for constructing
these surfaces. Are there other special surfaces supported by your CAD
software? Describe these special surfaces. ◇

Surface equations, like curve equations, can be classified into parametric representations and nonparametric representations. Consider a sphere of radius R centered at the origin of the reference coordinate system. The sphere can be expressed as

_{} (1)

The parametric representation of the sphere will be

_{},

_{} (2)

In general, two parameters are required to represent a surface,

_{}

_{} (3)

This parametric
equation uniquely maps the parametric space (_{} in *u* and *v* values) to the Cartesian space (_{} in *x*, *y*, and *z*). **To generate curves on a surface
patch, one can fix the value of one of the parametric variables, say u,
to obtain a curve in terms of the other variable, v.**

_{} (4)

Parametric representations of analytic surfaces of analytic surface are discussed as follows:

A plane defined
by three points _{} _{}, and _{} can be expressed
as

_{} _{} (5)

A ruled surface
is generated by joining corresponding points on two space curves (rails) **G**(*u*)
and **Q**(*u*) by straight lines (also called rulings or generators),
every developable surface is a ruled surface.

_{} (6)

_{} _{} (7)

A ruled surface
can only allow curvature in the *u* direction of the surface provided that
the rails have curvatures. **The surface curvature in the v direction (along the
rulings) is zero** and thus a ruled surface cannot be
used to model surface patches that have curvatures in two directions.

Assume the coordinates of control points of two cubic
Bezier curves **G**(*u*) and **Q**(*u*). Write a Matlab program
to construct a ruled surface using Equation (7). Display the surface by drawing
meshes for *v*=0, *v*=1, *u*=0, *u*=0.2, *u*=0.4, *u*=0.6,
*u*=0.8, *u*=1. Show your Matlab program too. ◇

For the surface you generated in Assignment 4, calculate
the tangent vectors at the point *u*=0.5, *v*=0.5. ◇

The rotation of
a planar curve an angle *v* about an axis of rotation creates a circle (if
*v*=360) for each point on the curve whose center lies on the axis of
rotation and whose radius _{} is variable, as
shown in Figure 11.** ****The planar curve and the circles are called the profile
and parallels respectively while the various positions of the profile around
the axis are called meridians**.

_{} _{} (8)

Figure 11. Parametric representation of a surface of revolution.

Assume the coordinates of the control points and create a
cubic Bezier curve _{}. Let _{}, _{}, and _{}, rewrite Equation (8). What is _{} in this case?
Write a Matlab program to plot this surface. Use proper mesh and view to
display the surface. Show your Matlab program too. ◇

A tabulated cylinder has been defined as a surface that results from translating a space planar curve along a given direction.

_{} _{} (9)