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## Coon's Patch Examples

Maple 13

Coon's patch defines a patch from 4 curves forming a chain. 4-examples to Coon's patch are given following worksheet. Boundary curves in the examples are constructed by parametric cubic curves (in Bezier and Hermite form). First example is in 3D and comparable to parametric bicubic surface. Rest of examples are in 2D (abstract surface) which can be described shortly as:

• 2D Example#1: 4 circular-arc like curves are combined by Coon's patch to form a circular-like region
• 2D Example#2: vertices of a square are assigned to rotating tangents
• 2D Example#3: boundaries corresponding to nozzle like geometry

 Introduction Coon's patch defines a surface by 4 curve segments that are connected in end-to-end to form curve chain. For left, right, bottom, top boundary curves denoted by  convex sum (blending) defines two lofting surfaces from  to  and  to  . First patch  interpolates  and second patch   interpolates  Adding two patches   and evaluating at boundaries yields   this shows that left boundary curve of   , which is sought to be has extra term that can be modelled by Expanding yields   from symmetry, there is no need to perform same procedure for other direction (~ bottom,right). The final form of patch to interpolate all four curves is
 Definitions

3D Coon's Patch Example

Define control points for 4 boundary curves

Define 4 Bezier curves

 (3.1)

plot corner curves

Plot control points for four corner curves

Define Coon's patch

 (3.2)

Plot defined Coon's patch

2D Coons Patch example-1

Define control points

Define corner curves

 (4.1)

Define Coon's patch

 (4.2)

plot 2D patch

plot corner curves

plot corner curves and surface

2D Coons Patch example-2 : square net with equally rotationg tangents

Define control points

Define corner curves

 (5.1)

Define Coon's patch

 (5.2)

a sample case

 (5.1.1)

 (5.1.2)

plot corner curves

plot corner curves and surface

plot 2D patch

animation for the sweep of tangent angle

2D Coon's Patch example-3 : grid generation for nozzle like geometry

Define control points

 (6.1)

Define corner curves

 (6.2)

Define Coon's patch

 (6.3)

plot 2D patch