# Tracing Geodesic Paths

The function traceGeodesic allows one to compute straightest paths along a surface (i.e. geodesic paths).

Note that straightest paths depend only on the intrinsic geometry of a surface (via the IntrinsicGeometryInterface). Therefore, these routines can be run on abstract geometric domains as well as traditional surfaces in 3D. However, these routines do assume that the domain is a ManifoldSurfaceMesh.

#include "geometrycentral/surface/trace_geodesic.h"

TraceGeodesicResult traceGeodesic(IntrinsicGeometryInterface& geom, SurfacePoint startP, Vector2 traceVec, const TraceOptions& traceOptions = defaultTraceOptions);

Trace a geodesic path along a surface mesh.

• inputGeom: the input geometry (as always, a VertexPositionGeometry is valid input)
• startP: the point on the surface where the path should start
• traceVec: the direction the path should proceed in, and the distance that it should travel
• traceOptions: options to specify the behavior of traceGeodesic in various situations

The function traceGeodesic traces out a geodesic path starting at startP which proceeds in the direction of traceVec and has length equal to traceVec.norm(), unless the path intersects a boundary edge in which case it stops there. This is also known as the exponential map. (As an aside, geometry-central also provides procedures for computing the inverse of the exponential map, known as the logarithmic map.)

### Example

#include "geometrycentral/surface/meshio.h"
#include "geometrycentral/surface/surface_point.h"
#include "geometrycentral/surface/trace_geodesic.h"

std::unique_ptr<ManifoldSurfaceMesh> mesh;
std::unique_ptr<VertexPositionGeometry> geometry;

Vertex v = mesh->vertex(0);
Vector2 traceVec = 3 * Vector2::fromAngle(M_PI/6);
SurfacePoint pathEndpoint = traceGeodesic(*geometry, SurfacePoint(v), traceVec).endPoint;


## Helper Types

### Options

Options are passed in to traceGeodesic via a TraceOptions object.

Field Default value Meaning
bool includePath false whether to return the entire path trajectory (as opposed to merely returning the path’s endpoint)
bool errorOnProblem false whether to throw exceptions if the procedure encounters degenerate geometry
EdgeData<bool>* barrierEdges nullptr if set, paths will stop when they hit barrier edges
size_t maxIters INVALID_IND if set, paths will stop after traversing through maxIters faces

### Result

The result is returned as a TraceGeodesicResult, which has 5 fields:

Field Meaning
SurfacePoint endPoint the point the path ended at
std::vector<SurfacePoint> pathPoints all points along the path, including start and end
Vector2 endingDir the incoming direction to the final point, in its tangent space
bool hitBoundary did the path stop early because we hit a boundary?
bool hasPath is pathPoints populated?

## Tangent Spaces

The input traceVec is specified as a vector in the tangent space of the starting point. The meaning of this vector depends on whether the starting point is located on a vertex, edge, or face of the mesh. Tangent space in geometry central are discussed in more detail on the Quantities page, but we give a brief overview here.

Given any mesh element (i.e. vertex, edge, or face) p, the $x$-axis of the tangent space at p points in the direction of p.halfedge(). The $y$-axis then points 90 degrees counterclockwise from the $x$-axis. (This is slightly more complicated at vertices, where one must use rescaled corner angles to define these directions. See the discussion of vertex tangent spaces) for more details.