Computer curve construction system

A computer curve construction method for constructing a curve that includes a plurality of connected curve components that are connected with a certain type of continuity, the method comprising:

• generating each curve component from a group of curves consisting of a peak-point curve, a point-point curve, a point-tangent curve, a point curve and a curvature curve;
• displaying the selected curve component;
• and
• connecting each subsequent curve component after a first curve component to a previous curve component with a particular type of continuity in a category consisting of geometric order 0 (G0), geometric order 1 (G1) and geometric order 2 (G2).

A computer curve construction system for constructing peak-point curves, the system comprising:

• an input device;
• and
• a processor coupled to the input device, and wherein the processor responds to signals received from the input device for:
• selecting a start point,
• selecting a start tangent direction at the start point,
• selecting an end point spaced from the start point,
• selecting an end tangent direction at the end point and
• selecting a peak point spaced from the start point and the end point,
• calculating a chord connecting the start point to the end point,
• the processor being operative to generate a first curve component between the start point and the end point and passing through the peak point having a tangent direction parallel to the chord connecting between the start point and the end point, the curve component being defined by a mathematical formula using the start point, the start tangent direction, the end point, the end tangent direction, the peak point and the peak point tangent direction as variables.

A computer curve construction method for constructing a peak-point curve, the method comprising:

• selecting a start point;
• selecting a start tangent direction at the start point;
• selecting an end point spaced from the start point;
• selecting an end tangent direction at the end point;
• calculating a direction of a chord connecting the start point and the end point;
• selecting a peak point spaced from the start point and the end point;
• and
• calculating a peak point tangent direction at the peak point parallel to the chord;
• generating a curve between the start point and the end point and passing through the peak point having a tangent direction parallel to that of the chord, the curve being defined by a mathematical formula using the start point, the start tangent direction, the end point, the end tangent direction, the peak point and the peak point tangent direction as variables.

A computer curve construction system for constructing point-point curves, the system comprising:

• an input device;
• and
• a processor coupled to the input device responding to signals received from the input device for:
• selecting a start point;
• selecting a start tangent direction;
• selecting an end point;
• and
• selecting a peak point between the start point and the end point,
• the processor calculating a chord vector connecting the start point and the end point;
• calculating an angle α between the start tangent direction and the chord vector;
• setting an end tangent direction at the end point intersecting the chord vector at angle β having a pre-selected relationship with angle α, and setting a peak point tangent direction at the peak point parallel to the chord vector,
• the processor being operative to generate a first curve between the start point and the end point and passing through the peak point, the curve being defined by a mathematical formula using the start point, the start tangent direction, the end point, the peak point and the peak point tangent direction as variables.

A computer curve construction method for constructing point-point curves, the method comprising:

• selecting a start point;
• selecting a start tangent direction at the start point;
• selecting an end point spaced from the start point;
• selecting a peak point spaced from the start point and from the end point;
• calculating a chord vector connecting the start point and the end point;
• calculating an angle a between the start tangent direction and the chord vector;
• setting an end tangent direction at the end point intersecting the chord vector at angle β where angle β has a pre-selected relationship with angle α;
• setting a peak point tangent direction at the peak point parallel to the chord vector;
• and generating a curve between the start point, the peak point and the end point, the curve being defined by a mathematical formula using the start point, the start tangent direction, the end point, the peak point and the peak point tangent direction as variables.

A computer curve construction system, comprising:

• an input device;
• and
• a processor coupled to the input device, wherein the processor responds to signals
• received from the input device for:
• selecting a start point;
• establishing a start tangent direction;
• selecting an end point;
• the processor:
• calculating a chord between the start point and the end point;
• calculating an angle α between the start tangent direction and the chord;
• setting an end tangent direction at the end point at an angle β related to angle α, and
• calculating a peak point having a tangent direction parallel to the chord located on a line bisecting the chord and extending through a vertex point defined by an intersection of a first ray aligned with the start tangent direction and a second ray aligned with the end tangent direction,
• the processor being operative to generate a first curve defined by a mathematical formula between the start point and the end point and passing through the peak point using the start point, the start tangent direction, the end point, the end tangent direction, the peak point and the peak point tangent direction as variables.

A computer curve construction method for constructing a curve, the method comprising:

• selecting a start point;
• selecting a start tangent direction at the start point;
• selecting an end point spaced from the start point,
• calculating a first angle α between the start tangent direction and a chord vector between the start point and the end point;
• setting an end tangent direction at the end point at a second angle β to the chord vector related to the first angle α;
• locating a peak point having a tangent direction parallel to the chord vector on a line bisecting the chord vector and extending through a vertex point defined by an intersection of a first ray coinciding and aligned with the start tangent direction and a second ray coinciding and aligned with the end tangent direction;
• and
• generating a first curve defined by a mathematical formula between the start point and the end point and passing through the peak point using the start point, the start tangent direction, the end point, the end point tangent direction, the peak point and the peak point tangent direction as variables.

A computer curve construction method for constructing a curvature curve, the method comprising:

• selecting a start point,
• selecting a start tangent direction at the start point,
• calculating and displaying a first locus line that includes the start point and is perpendicular to the start tangent direction,
• selecting a start curvature at the start point by selecting a center of a start curvature circle that includes the start point along the first locus line,
• selecting an end point spaced from the start point,
• selecting an end tangent direction at the end point,
• calculating and displaying a second locus line that is a ray or a line segment on the line that includes the end point and is perpendicular to the end tangent direction,
• selecting an end curvature at the end point by selecting a center of an end curvature circle that includes the end point along the second locus line, and
• generating and displaying a first curve component between the start point and the end point, the curve being defined by mathematical formula using the start point, the start tangent direction, the start curvature, the end point, the end tangent direction and the end curvature as variables.

A computer curve construction system for constructing curvature curves comprising, in combination:

• an input device;
• a display device for presenting two dimensional representations, and
• a processor receiving input from the input device, and providing output driving the display device,
• wherein the processor responds to signals received from the input device for:
• selecting a start point,
• selecting a start tangent direction at the start point,
• specifying a start curvature at the start point,
• selecting an end point spaced from the start point,
• selecting an end tangent direction at the end point and
• specifying an end curvature at the end point,
• wherein the start curvature at the start point is specified by an operator using the input device to establish a start curvature circle that includes the start point by selecting a center with the input device along a first displayed locus line that includes the start point and is perpendicular to the start tangent direction, and
• wherein the end curvature at the end point is specified by an operator using the input device to establish an end curvature circle that includes the end point by selecting a center with the input device along second displayed locus line segment that includes the end point and is perpendicular to the end tangent direction,
• the processor being operative to generate and provide output to the display device for displaying a curve between the start point and the end point, the curve being defined by mathematical formula using the start point, the start tangent direction, the start curvature, the end point, the end tangent direction and the end curvature as variables.

A computer curve construction system for constructing curves in a class consisting of a peak-point curve, a point-point curve, a point-tangent curve, and a point curve, comprising, in combination:

• an input device;
• a display device for presenting two dimensional representations, and
• a processor coupled to the input device, wherein the processor responds to signals received from the input device for
• selecting a start point a0;
• establishing a start tangent direction unit vector e0 at the start point;
• selecting an end point a1;
• establishing an end tangent direction unit vector e1 at the end point where signature(e0×(a1−a0))=signature(e1×(a0−a1))≠0,;
• the processor providing output to the display device for indicating a two dimensional region bounded by a first ray coinciding with and extending in the start tangent direction unit vector e0, a chord connecting a0 and a1, and a second ray coinciding with and extending opposite to the end tangent direction unit vector e1, where a computer graphics operator can establish a peak point for the curve having a tangent direction parallel to the chord connecting a0 and a1, the processor providing output to the display device for displaying the curve upon operator establishment of the peak point.

A computer curve construction method for constructing curves in a class consisting of a peak-point curve, a point-point curve, a point-tangent, and a point curve comprising the steps of:

• a) selecting a start point a0;
• b) selecting a start tangent direction unit vector e0 at the start point;
• c) selecting an end point a1;
• d) establishing an end tangent direction unit vector e1 at the end point where signature(e0×(a1−a0))=signature(e1×(a0−a1))≠0;
• e) calculating a chord connecting a0 and a1;
• f) displaying a two dimensional region bounded by a first ray coinciding and extending with the start tangent direction unit vector e0, a chord connecting a0 and a1, and a second ray coinciding with and extending opposite to the end tangent direction unit vector e1;
• g) locating a peak point within the displayed two dimensional region;
• h) calculating a peak point tangent direction at the peak point parallel the chord connecting a0 and a1;
• i) displaying a first curve between the start point and the end point, passing through the peak point, the curve being defined by a mathematical formula using the start point, the start tangent direction, the end point, the end tangent direction, the peak point and the peak point tangent direction as variables.

A computer curve construction system for constructing a curvature curve, the system comprising:

• an input device,
• a display device for presenting two-dimensional representations, and
• a processor coupled to the input device providing output driving the display device,
• wherein the processor responds to selection signals received from the input device specifying and establishing geometric constraints for the constructed cure by
• selecting a start point a0,
• selecting a start tangent direction unit vector e0 at the start point,
• establishing a start curvature k0 at the start point by
• selection of a center of curvature circle on an imaginary locus line that includes the start point and is perpendicular to the start tangent direction indicated by the display device from output of the processor,
• selecting an end point a1,
• selecting an end tangent direction unit vector e1 at the end point where:
• signature(e0×(a1−a0))=signature(e1×(a0−a1))≠0 and
• signature(k0)=signature(e0×(a1−a0)) if k0≠0, and
• signature(e0×e1)=signature(e0×(a1−a0)) if k0=0,
• establishing an end curvature k1 at the end point by
• selection of a center of end curvature circle on an imaginary locus line segment on the line that includes the end point and is perpendicular to the end tangent direction indicated by the display device from output of the processor,
• the processor being operative to generate output to the display device for displaying a first curve between the start point and the end point, the curve being defined by a mathematical formula using the selected and established geometric constraints as variables.

A computer curve construction method for constructing a curvature curve by selecting and establishing geometric constraints for constructing the method comprising:

• a) selecting a start point a0 with an input device coupled to a CPU processor,
• b) selecting a start tangent direction unit vector e0 at the start point with the input device,
• c) displaying a first imaginary locus line that includes the start point and is perpendicular to the start tangent direction with a display device driven by output from the CPU processor;
• d) establishing a start curvature at the start point by selecting a center of start curvature circle on the displayed first imaginary locus line with the input device;
• e) selecting an end point a1 with the input device,
• f) selecting an end tangent direction unit vector e1 at the end point with the input device where:
• signature(e0×(a1−a0))=signature(e1×(a0−a1))≠0, and
• signature(k0)=signature(e0×(a1−a0)) if k0≠0, and
• signature(e0×e1)=signature(e0×(a1−a0)) if k0=0,
• g) displaying a second imaginary locus line segment that includes the end point and is perpendicular to the end tangent direction with the display device driven by output from the CPU processor;
• h) establishing an end curvature at the end point by selecting a center of end curvature circle on the displayed second imaginary locus line segment with the input device,
• i) displaying a first curve between the start point and the end point with the display device, the curve being generated from output of the CPU processor using a mathematical formula and the selected and established geometric constraints as variables.