# Parallel Curve Algorithms

Parallel curve algorithms, also known as offset algorithms, are computational methods used in computer graphics and computational geometry to generate curves that run in parallel to an existing curve or set of curves. These algorithms are employed in various applications, such as offsetting boundaries, generating tool paths for machining, and creating visual effects in computer-aided design (CAD) and computer-aided manufacturing (CAM) systems.

The primary goal of parallel curve algorithms is to construct a new curve that maintains a consistent offset distance from the original curve(s) while preserving their overall shape and topology. The offset distance can be positive (outside) or negative (inside) relative to the original curve(s). The resulting parallel curve(s) can be used to represent boundaries for tool clearance, generate concentric shapes, or create aesthetically pleasing visual effects.

Several parallel curve algorithms exist, each with its own characteristics, advantages, and limitations. Here are some commonly used techniques:

1. Offset Curves by Arcs: This algorithm approximates the parallel curve by a sequence of circular arcs, each tangent to adjacent arcs. It works well for curves with relatively low curvature but may introduce noticeable errors for sharp corners or high-curvature sections.
2. Offset Curves by Straight Segments: This method approximates the parallel curve using a series of line segments, each connected to the neighboring segments. It is relatively simple to implement and works for both open and closed curves. However, it may produce jagged results for curves with high curvature or small offset distances.
3. Medial Axis Transform (MAT): The MAT algorithm constructs the skeleton or medial axis of the original curve(s), which represents the set of all points having more than one closest point on the curve(s). By dilating the skeleton by the desired offset distance, a parallel curve is obtained. MAT can handle complex and self-intersecting curves but may introduce topological changes.
4. Discrete Curve Evolution: This approach discretizes the original curve(s) into a set of points and evolves these points along the normal direction to generate the parallel curve(s). It can handle curves with arbitrary shapes and sharp corners but may require careful handling of self-intersections and degenerate cases.
5. Biarc Approximation: Biarc approximation algorithms represent the parallel curve(s) using a sequence of circular arcs and line segments. They aim to find the best-fitting sequence of arcs and lines that match the offset distance. Biarc approximations are often used in computer-aided manufacturing to generate tool paths with smooth transitions.

## Offset Curves by Arcs Algorithm

The Offset Curves by Arcs algorithm is a method used to generate parallel curves by approximating them with a sequence of circular arcs. This algorithm is commonly employed in computer graphics and computational geometry to offset curves while preserving their overall shape.

The basic idea of the Offset Curves by Arcs algorithm is to compute a sequence of circular arcs that are tangent to the original curve and maintain a consistent offset distance. The algorithm works by iteratively constructing arcs that connect the endpoints of each line segment on the original curve.

Here is a step-by-step overview of the Offset Curves by Arcs algorithm:

1. Given an original curve and the desired offset distance, select a starting point on the curve.
2. Move along the curve in one direction (either clockwise or counterclockwise) and divide the curve into line segments.
3. For each line segment, determine the radius of the circular arc that will approximate the offset curve. This is typically done by computing the perpendicular bisector of the line segment and selecting a radius based on the offset distance.
4. Construct a circular arc using the computed radius, starting at the endpoint of the previous arc or the starting point of the curve.
5. Continue constructing arcs for each line segment, ensuring that they are tangent to each other. The tangency condition guarantees a smooth transition between the arcs, preserving the overall shape of the offset curve.
6. Once all line segments have been processed, connect the endpoint of the last arc to the starting point of the curve to complete the offset curve.

It’s important to note that the Offset Curves by Arcs algorithm has some limitations. It works well for curves with low curvature or gradual changes in curvature but may introduce noticeable errors for curves with sharp corners or high-curvature sections. In such cases, additional techniques or algorithms may be necessary to achieve accurate results.

The Offset Curves by the Arcs algorithm are just one of several methods available for generating parallel curves. The choice of algorithm depends on the specific requirements of the application, such as the desired accuracy, efficiency, and the characteristics of the original curve.

## Offset Curves by Straight SegmentsAlgorithm

The Offset Curves by Straight Segments algorithm is a technique used to generate parallel curves by approximating them with a series of straight line segments. This algorithm is commonly employed in computer graphics and computational geometry to offset curves while preserving their overall shape.

The Offset Curves by Straight Segments algorithm works by dividing the original curve into smaller line segments and then offsetting each segment by a consistent distance. Here’s a step-by-step overview of the algorithm:

1. Given an original curve and the desired offset distance, divide the curve into smaller line segments. The number of segments and their lengths depend on the desired accuracy and the curvature of the original curve.
2. For each line segment, compute the direction perpendicular to the segment. This direction represents the offset direction for the new curve.
3. Move each endpoint of the line segment along the offset direction by the specified offset distance. This creates two new points that are offset from the original endpoints.
4. Connect the offset endpoints of adjacent line segments with straight line segments. These new line segments approximate the parallel curve.
5. Repeat steps 2 to 4 for all line segments in the original curve.
6. If the original curve is closed, connect the endpoints of the last line segment to the starting point to close the parallel curve.

The Offset Curves by Straight Segments algorithm is relatively simple to implement and works well for both open and closed curves. However, it may produce jagged results for curves with high curvature or small offset distances. To improve the smoothness of the parallel curve, more line segments can be added, or additional algorithms can be employed to refine the offset.

It’s worth noting that there are other techniques available for generating parallel curves, each with its own advantages and limitations. The choice of algorithm depends on the specific requirements of the application, such as the desired accuracy, efficiency, and the characteristics of the original curve.

## Medial Axis Transform (MAT)Algorithm

The Medial Axis Transform (MAT) algorithm is a computational technique used in computer graphics and computational geometry to generate parallel curves by constructing the skeleton or medial axis of the original curve(s). This algorithm is commonly employed to offset curves while preserving their overall shape and topology.

The Medial Axis Transform algorithm works by computing the set of all points in the plane that have more than one closest point on the original curve(s). These points represent the skeleton or medial axis, which captures the centerlines or core structure of the curve(s). By dilating the skeleton by the desired offset distance, a parallel curve is obtained.

Here’s a step-by-step overview of the Medial Axis Transform algorithm:

1. Given the original curve(s) and the desired offset distance, compute the skeleton or medial axis. There are various techniques to calculate the medial axis, such as Voronoi diagrams, distance transforms, or curve skeletonization algorithms.
2. Dilate the skeleton by the offset distance. This is done by expanding each point on the skeleton outward in all directions by the specified distance.
3. Connect neighboring dilated points to form the parallel curve(s). The resulting curve(s) will maintain a consistent offset distance from the original curve(s) while preserving their overall shape.

The Medial Axis Transform algorithm can handle curves with complex shapes, self-intersections, and holes. It provides a robust approach to generating parallel curves that adapt to the geometry of the original curve(s). However, it’s important to note that the algorithm may introduce topological changes, such as merging or splitting of components, depending on the characteristics of the original curve(s).

The Medial Axis Transform algorithm is just one of several methods available for generating parallel curves. The choice of algorithm depends on the specific requirements of the application, such as the desired accuracy, efficiency, and the characteristics of the original curve(s).

## Discrete Curve EvolutionAlgorithm

The Discrete Curve Evolution algorithm is a computational technique used in computer graphics and computational geometry to generate parallel curves by evolving a discrete set of points along the normal direction of the original curve(s). This algorithm is commonly employed to offset curves while preserving their overall shape and accommodating arbitrary curve shapes, including sharp corners.

The Discrete Curve Evolution algorithm works by discretizing the original curve(s) into a set of points and iteratively evolving these points to generate the parallel curve(s). Here’s a step-by-step overview of the algorithm:

1. Given the original curve(s) and the desired offset distance, discretize the curve(s) into a set of evenly spaced points. The number of points depends on the desired accuracy and the curvature of the original curve(s).
2. For each point on the original curve(s), compute the normal direction. The normal direction represents the direction in which the point will be offset to generate the parallel curve(s).
3. Move each point along its normal direction by the specified offset distance. This displacement creates a new set of points that are offset from the original curve(s).
4. Connect neighboring offset points with straight line segments. These line segments approximate the parallel curve.
5. Repeat steps 2 to 4 for a sufficient number of iterations or until convergence is achieved. In each iteration, update the normal directions based on the new positions of the offset points to ensure smooth transitions and accurate offsets.
6. If the original curve is closed, connect the endpoints of the last line segment to the starting point to close the parallel curve.

The Discrete Curve Evolution algorithm can handle curves with arbitrary shapes, including sharp corners and intricate details. It offers flexibility in offsetting curves while preserving their local and global characteristics. However, special care must be taken to handle self-intersections and degenerate cases that may arise during the offset process.

It’s important to note that the Discrete Curve Evolution algorithm is just one of several methods available for generating parallel curves. The choice of algorithm depends on the specific requirements of the application, such as the desired accuracy, efficiency, and the characteristics of the original curve(s).

## Biarc ApproximationAlgorithm

The Biarc Approximation algorithm is a computational technique used in computer graphics and computational geometry to generate parallel curves by approximating them with a sequence of circular arcs and line segments. This algorithm is commonly employed to offset curves while ensuring smooth transitions and maintaining the overall shape of the original curve(s).

The Biarc Approximation algorithm works by finding the best-fitting sequence of circular arcs and line segments that match the desired offset distance. Here’s a step-by-step overview of the algorithm:

1. Given the original curve(s) and the desired offset distance, select a starting point on the curve.
2. Move along the curve in one direction (either clockwise or counterclockwise) and divide the curve into smaller line segments.
3. For each line segment, attempt to fit a circular arc that matches the offset distance. The arc should be tangent to both the line segment and the adjacent line segments.
4. If a suitable arc cannot be found, approximate the offset by a line segment connecting the endpoints of the original line segment.
5. Connect the endpoints of the arcs and line segments to form the parallel curve(s). These arcs and line segments approximate the offset curve.
6. Repeat steps 2 to 5 for all line segments in the original curve.
7. If the original curve is closed, connect the endpoints of the last arc or line segment to the starting point to close the parallel curve.

The Biarc Approximation algorithm aims to create a smooth offset curve by using circular arcs wherever possible. However, if the original curve has sharp corners or high-curvature sections, the algorithm may resort to line segments to maintain accuracy. This combination of arcs and line segments results in an approximation of the parallel curve(s) that balances smoothness and fidelity to the original shape.

The Biarc Approximation algorithm is often used in computer-aided manufacturing to generate tool paths with smooth transitions. By using a combination of arcs and line segments, it achieves efficient and visually pleasing offset curves while taking into account the constraints of the original curve(s) and the desired offset distance.

It’s important to note that the Biarc Approximation algorithm is just one of several methods available for generating parallel curves. The choice of algorithm depends on the specific requirements of the application, such as the desired accuracy, efficiency, and the characteristics of the original curve(s).

Here’s an example of how you could implement a parallel curve algorithm in the C programming language:

#include <stdio.h>
#include <math.h>

// Function to compute the parallel curve of a given curve
void computeParallelCurve(double *originalCurveX, double *originalCurveY, int numPoints, double offsetDistance,
double *parallelCurveX, double *parallelCurveY) {
int i;

// Compute the normal vector for each point on the original curve and normalize it
for (i = 0; i < numPoints; i++) {
double dx = originalCurveX[i + 1] - originalCurveX[i]; // x-component of the tangent vector
double dy = originalCurveY[i + 1] - originalCurveY[i]; // y-component of the tangent vector

double length = sqrt(dx * dx + dy * dy); // length of the tangent vector
double nx = -dy / length; // x-component of the normalized normal vector
double ny = dx / length;  // y-component of the normalized normal vector

// Compute the offset point using the normalized normal vector and the offset distance
parallelCurveX[i] = originalCurveX[i] + offsetDistance * nx;
parallelCurveY[i] = originalCurveY[i] + offsetDistance * ny;
}
}

int main() {
// Example usage
double originalCurveX[] = {0.0, 1.0, 2.0, 3.0}; // X-coordinates of the original curve points
double originalCurveY[] = {0.0, 1.0, 0.5, 0.0}; // Y-coordinates of the original curve points
int numPoints = sizeof(originalCurveX) / sizeof(originalCurveX[0]);

double offsetDistance = 0.5; // Desired offset distance

// Allocate memory for the parallel curve points
double *parallelCurveX = (double *) malloc(numPoints * sizeof(double));
double *parallelCurveY = (double *) malloc(numPoints * sizeof(double));

// Compute the parallel curve
computeParallelCurve(originalCurveX, originalCurveY, numPoints, offsetDistance, parallelCurveX, parallelCurveY);

// Print the coordinates of the parallel curve points
for (int i = 0; i < numPoints; i++) {
printf("Point %d: (%lf, %lf)\n", i, parallelCurveX[i], parallelCurveY[i]);
}

// Free the allocated memory
free(parallelCurveX);
free(parallelCurveY);

return 0;
}