Method and system for inversion of detail-in-context presentations

1. In a data processing system that executes a program of instructions, a method for inverting a point X on a distorted surface in a detail-in-context presentation for display on a display screen, comprising the steps of:

(a) locating a first approximation point P i , for an inversion of the point X, wherein the point P i is on an undistorted surface; and,

(b) obtaining a point P i D by displacing the point P i , onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference|P i D −X| between the point X and the point P i D ; determining whether the point P i is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ; and, displaying the point P i on the display screen if the point P i is acceptable for the inversion of the point X.

2. The method of claim 1 and further comprising the steps of:

(c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i is not acceptable for the inversion of the point X; and,

(d) repeating steps (b) and (c) until the approximation point is acceptable for the inversion of the point X.

3. The method of claim 2 and further comprising the step of selecting the tolerance δ.

4. The method of claim 3 wherein the tolerance δ is a fraction of a width of a pixel for a computer display surface.

5. The method of claim 4 wherein the fraction includes one-half.

6. The method of claim 1 wherein the undistorted surface is included in the detail-in-context presentation.

7. The method of claim 2 and further comprising the step of constructing a line RVP-X from a point RVP above the undistorted surface, through the point X, and through the undistorted surface to locate the first approximation point P i at a point of intersection of the line RVP-X and the undistorted surface.

8. The method of claim 7 wherein the point RVP is a reference viewpoint for the detail-in-context presentation.

9. The method of claim 8 and further comprising the steps of:

projecting the point P i D onto the line RVP-X to locate a point P i P , wherein the point P i P , is a closest point to the point P i D on the line RVP-X; and,

projecting the point P i P onto the undistorted surface in a direction opposite to that of a displacement due to distortion to locate the next approximation point P i+1 for the inversion of the point X, wherein the displacement due to distortion is given by a line F o - F constructed through the undistorted surface and a focus F of the distorted surface, and wherein the point P i+1 is located on the undistorted surface at a point of intersection of the undistorted surface and a line constructed parallel to the line F o - F and passing through the point P i P .

10. The method of claim 2 and further comprising the step of bisecting the point P i to counter divergence in successive approximations of the point P i due to folds or discontinuities in the distorted surface.

11. The method of claim 1 wherein the undistorted surface is a plane.

12. The method of claim 1 wherein the distorted surface is defined by the distortion function D.

13. The method of claim 12 wherein the distortion function D is an n-dimensional function, wherein n is an integer greater than zero.

14. The method of claim 13 wherein the distortion function D is a three-dimensional function.

15. The method of claim 12 wherein the distortion function D is a lens function.

16. A system for inverting a point X on a distorted surface in a detail-in-context presentation, the system having memory, a display, and an input device, the system comprising:

a processor coupled to the memory, display, and input device and adapted for:

(a) locating a first approximation point P i for an inversion of the point X, wherein the point P i is on an undistorted surface; and,

(b) obtaining a point P i D by displacing the point P i onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i D −X| between the point X and the point P i D ; determining whether the point P i is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ; and, displaying the point P i on the display if the point P i , is acceptable for the inversion of the point X.

17. The system of claim 16 wherein said processor is further adapted for:

(c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i is not acceptable for the inversion of the point X; and,

(d) repeating (b) and (c) until the approximation point is acceptable for the inversion of the point X.

18. A computer program product having a computer readable medium tangibly embodying computer executable code for directing a data processing system to invert a point X on a distorted surface in a detail-in-context presentation for display on a display screen , the computer program product comprising:

code for (a) locating a first approximation point P i for an inversion of the point X, wherein the point P i is on an undistorted surface; and,

code for (b) obtaining a point P i D by displacing the point P i onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i D −X| between the point X and the point P i D ; determining whether the point P i , is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ; and, displaying the point P i on the display screen if the point P i is acceptable for the inversion of the point X.

19. The computer program product of claim 18 and further comprising:

code for (c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i is not acceptable for the inversion of the point X; and,

code for (d) repeating (b) and (c) until the approximation point is acceptable for the inversion of the point X.

20. In a data processing system that executes a program of instructions, a method for determining a distance on an undistorted surface between a first point X 1 and a second point X 2 on a distorted surface in a detail-in-context presentation for display on a display screen, comprising:

inverting the point X 1 by:

locating a first approximation point P i1 for an inversion of the point X 1 , wherein the point P i1 is on the undistorted surface; and,

obtaining a point P i1 D by displacing the point P i1 onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i1 D−X 1 | between the point X 1 and the point P i1 D ; and, determining whether the point P i1 is acceptable for the inversion of the point X 1 by comparing the magnitude of the difference |P i1 D −X 1 | to a tolerance δ;

inverting the point X 2 by:

locating a first approximation point P i2 for an inversion of the point X 2 , wherein the point P i2 is on the undistorted surface; and,

obtaining a point P i2 D by displacing the point P i2 onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i2 D −X 2 | between the point X 2 and the point P i2 D ; and, determining whether the point P i2 is acceptable for the inversion of the point X 2 by comparing the magnitude of the difference |P i2 D −X 2 | to the tolerance δ;

calculating a magnitude of the difference |P i1 −P i2 | between the approximation points P i1 and P i2 ; and,

displaying the maanitude of the difference |P i1 −P i2 | on the display screen.

21. The method of claim 20 wherein the first point X 1 is on a first disported surface defined by a first distortion function D 1 and the second point X 2 is on a second distorted surface defined by a second distortion function D 2 .

22. In a data processing system that executes a program of instructions, a method for inverting a point X on a distorted surface in a detail-in-context presentation for display on a display screen, comprising the steps of:

(a) locating a first approximation point P i for an inversion of the point X, wherein the point P i is on an undistorted surface;

(b) obtaining a point P i D by displacing the point P i onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i D −X| between the point X and the point P i D ; and, detennining whether the point P i is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ;

(c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i , is not acceptable for the inversion of the point X by: constructing a line RVP-X from a point RVP above the undistorted surface, through the point X, and through the undistorted surface to locate the first approximation point P i at a point of intersection of the line RVP-X and the undistorted surface, wherein the point RVP is a reference viewpoint for the detail-in-context presentation; projecting the point P i D onto the line RVP-X to locate a point P i P , wherein the point P i P is a closest point to the point P i D on the line RVP-X; and, projecting the point P i P onto the undistorted surface in a direction opposite to that of a displacement due to distortion to locate the next approximation point P i+1 for the inversion of the point X, wherein the displacement due to distortion is given by a line F o - F constructed through the undistorted surface and a focus F of the distorted surface, and wherein the point P i+1 is located on the undistorted surface at a point of intersection of the undistorted surface and a line constructed parallel to the line F o - F and passing through the point P i P ;

(d) repeating steps (b) and (c) until the approximation point is acceptable for the inversion of the point X; and,

(e) displaying the approximation point on the display screen.

23. A computer program product having a computer readable medium tangibly embodying computer executable code for directing a data processing system to invert a point X on a distorted surface in a detail-in-context presentation for display on a display screen , the computer program product comprising:

code for (a) locating a first approximation point P i for an inversion of the point X, wherein the point P i is on an undistorted surface;

code for (b) obtaining a point P i D by displacing the point P i onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i D −X| between the point X and the point P i D ; and, determining whether the point P i is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ;

code for (c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i is not acceptable for the inversion of the point X;

code for (d) repeating (b) and (c) until the approximation point is acceptable for the inversion of the point X; and,

code for (e) displaying the approximation point on the display screen.

24. In a data processing system that executes a program of instructions, a method for inverting a point X on a distorted surface in a detail-in-context presentation for display on a display screen, comprising the steps of:

(a) locating a first approximation point P i for an inversion of the point X, wherein the point P i is on an undistorted surface;

(b) obtaining a point P i D by displacing the point P i onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i D −X| between the point X and the point P i D ; and, determining whether the point P 1 is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ;

(c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i is not acceptable for the inversion of the point X; and,

(d) repeating steps (b) and (c) until the approximation point is acceptable for the inversion of the point X; and,

(e) displaying the approximation point on the display screen.

25. A system for inverting a point X on a distorted surface in a detail-in-context context presentation, the system having memory, a display, and an input device, the system comprising:

a processor coupled to the memory, display, and input device and adapted for:

(a) locating a first approximation point P i for an inversion of the point X, wherein the point P i is on an undistorted surface;

(b) obtaining a point P i D by displacing the point P i onto the distorted surface by applying a distortion function D; calculating a magnitude of the difference |P i D −X| between the point X and the point P i D ; and, determining whether the point P i is acceptable for the inversion of the point X by comparing the magnitude of the difference to a tolerance δ;

(c) locating a next approximation point P i+1 for the inversion of the point X if the approximation point P i is not acceptable for the inversion of the point X; and,

(d) repeating (b) and (c) until the approximation point is acceptable for the inversion of the point X; and,

(e) displaying the approximation point on the display.