Method for evaluating error in shape of free curved surface

• US 6,711,530 B1
• Filed: 04/18/2000
• Issued: 03/23/2004
• Est. Priority Date: 08/19/1998
• Status: Expired due to Fees

• Current Assignee: Riken
• Original Assignee: Riken
• Inventors: Kase, Kiwamu
• Primary Examiner(s): Frejd, Russell
• Assistant Examiner(s): Craig, Dwin M.
• Application Number:
  US09/529,663
• Time in Patent Office: 1,435 Days
• Field of Search: 703/2, 382/242, 382/266
• US Class Current: 703/2
• CPC Class Codes: G01B 21/20 for measuring contours or curvatures, e.g. determining profile
• CPC Class Codes: G06T 17/30 Polynomial surface description

Abstract

An original curved surface S is divided into up to six curved surface units by combinations of signs (+, 0, −) of a principal curvature (k₁, k₂) in each point on the curved surface. A distorted curved surface S′ is associated with the original curved surface S and divided into curved surface units having the same boundary. An average normal vector is obtained for each curved surface unit with respect to the original curved surface and the distorted curved surface. “A bent component” and “a twisted component” in all the combinations of pairs of different curved surface units are obtained with respect to the original curved surface and the distorted curved surface. A difference between “the bent component” and “the twisted component” in each combination in the original curved surface and “the bent component” and “the twisted component” in each identical combination in the distorted curved surface is calculated.

Claims

1. A method for evaluating a shape error of a free curved surface comprising:

dividing an original curved surface S into up to six curved surface units by combinations of signs (+, 0, −

) of a principal curvature (k₁, k₂) in each point on said curved surface;

associating a distorted curved surface S′

with said original curved surface S and dividing it into curved surface units having the same boundary by projection along normal vectors of the curved surface S;

calculating an average normal vector for each curved surface unit with respect to said original curved surface and said distorted curved surface;

obtaining “

a bent component” and

“

a twisted component”

in all the combinations of pairs of different curved surface units with respect to said original curved surface and said distorted curved surface; and

calculating a difference between “

the bent component” and

“

the twisted component”

in each combination in said original curved surface and “

the bent component” and

“

the twisted component”

in each identical combination in said distorted curved surface.

2. A method for evaluating a shape error of a free curved surface comprising the steps of:

dividing an original curved surface S into up to six curved surface units by combinations of signs (+, 0, −

) of a principal curvature (K₁, K₂) in each point on the curved surface S;

associating a distorted curved surface S′

with the original curved surface S and dividing the distorted curved surface S′

into curved surface units having the same boundary by projection along normal vectors of the curved surface S;

calculating an average normal vector for each curved surface unit with respect to the original curved surface S and the distorted curved surface S′

, wherein each curved surface unit is divided into two regions having orthogonal directions and average normal vectors in the two divided regions are calculated;

obtaining a bent component and a twisted component in all combinations of pairs of different curved surface units with respect to the original curved surface S and the distorted curved surface S′

, wherein the bent component and the twisted component are obtained with respect to combinations of pairs of the average normal vectors in the two divided regions in each curved surface unit; and

calculating a difference between the bent component and the twisted component in each combination in the original curved surface S and the bent component and the twisted component in each identical combination in the distorted curved surface S′

.

3. A method for evaluating a shape error of a free curved surface according to claim 2, wherein a bent component θ

is calculated by Expression 1;

Expression 1 $\begin{array}{cc}\mathrm{sgn}<br><br>\left(\stackrel{\to <br><br>}{\mathrm{AB}},\stackrel{\to <br><br>}{\mathrm{CD}},\stackrel{\to <br><br>}{\mathrm{CA}}\right)=\{\begin{array}{cc}+1& \mathrm{if}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\stackrel{\to <br><br>}{\mathrm{AB}}-\stackrel{\to <br><br>}{\mathrm{CD}}\right)·<br><br>\stackrel{\to <br><br>}{\mathrm{CA}}><br><br>0\\ -1& \mathrm{else}\end{array},& \left(3\right)\\ \stackrel{\to <br><br>}{n}=\frac{\stackrel{\to <br><br>}{\mathrm{AB}}\times <br><br>\stackrel{\to <br><br>}{\mathrm{AC}}}{<br><br>\stackrel{\to <br><br>}{\mathrm{AB}}\times <br><br>\stackrel{\to <br><br>}{\mathrm{AC}}<br><br>}& \left(4\right)\\ \stackrel{\to <br><br>}{{\mathrm{CD}}^{\mathrm{\prime <br><br>}}}=\stackrel{\to <br><br>}{\mathrm{CD}}-\left(\stackrel{\to <br><br>}{n}·<br><br>\stackrel{\to <br><br>}{\mathrm{CD}}\right)<br><br>\stackrel{\to <br><br>}{n}& \left(5\right)\\ \mathrm{\theta <br><br>}=\mathrm{sgn}\left(\stackrel{\to <br><br>}{\mathrm{AB}},\stackrel{\to <br><br>}{\mathrm{CD}},\stackrel{\to <br><br>}{\mathrm{CA}}\right)<br><br><br><br>\mathrm{Arccos}<br><br>\left(\frac{\stackrel{\to <br><br>}{\mathrm{AB}}·<br><br>\stackrel{\to <br><br>}{{\mathrm{CD}}^{\mathrm{\prime <br><br>}}}}{<br><br>\stackrel{\to <br><br>}{\mathrm{AB}}·<br><br>\stackrel{\to <br><br>}{{\mathrm{CD}}^{\mathrm{\prime <br><br>}}}<br><br>}\right)<br><br>,& \left(6\right)\end{array}$

and a twisted component φ

can be calculated by Expression 2;

Expression 2$\begin{array}{cc}\mathrm{\phi <br><br>}=\mathrm{Arcsin}<br><br>\left(\frac{\left(\stackrel{\to <br><br>}{\mathrm{AC}}\times <br><br>\left(\stackrel{\to <br><br>}{\mathrm{AB}}\times <br><br>\stackrel{\to <br><br>}{\mathrm{AC}}\right)\right)\times <br><br>\left(\stackrel{\to <br><br>}{\mathrm{AC}}\times <br><br>\left(\stackrel{\to <br><br>}{\mathrm{CD}}\times <br><br>\stackrel{\to <br><br>}{\mathrm{AC}}\right)\right)}{<br><br>\left(\stackrel{\to <br><br>}{\mathrm{AC}}\times <br><br>\left(\stackrel{\to <br><br>}{\mathrm{AB}}\times <br><br>\stackrel{\to <br><br>}{\mathrm{AC}}\right)\right)<br><br><br><br><br><br><br><br>\left(\stackrel{\to <br><br>}{\mathrm{AC}}\times <br><br>\left(\stackrel{\to <br><br>}{\mathrm{CD}}\times <br><br>\stackrel{\to <br><br>}{\mathrm{AC}}\right)\right)<br><br>}·<br><br>\frac{\stackrel{\to <br><br>}{\mathrm{AC}}}{<br><br>\stackrel{\to <br><br>}{\mathrm{AC}}<br><br>}\right)& \left(7\right)\end{array}$

where {right arrow over (AB)}×

{right arrow over (CD)} represents an outer product of the vectors of AB and CD.