Classes
struct	RayIntersection
	Result of ray-ellipsoid intersection. More...

Functions
template<typename Scalar>
Scalar	haversine_distance (const LLA< Scalar > &lla1, const LLA< Scalar > &lla2, double radius=constants::earth::R_mean)
template<typename Scalar>
Scalar	great_circle_distance (const LLA< Scalar > &lla1, const LLA< Scalar > &lla2, const EarthModel &m=EarthModel::WGS84())
template<typename Scalar>
Scalar	initial_bearing (const LLA< Scalar > &lla1, const LLA< Scalar > &lla2, const EarthModel &m=EarthModel::WGS84())
template<typename Scalar>
Scalar	final_bearing (const LLA< Scalar > &lla1, const LLA< Scalar > &lla2, const EarthModel &m=EarthModel::WGS84())
template<typename Scalar>
LLA< Scalar >	destination_point (const LLA< Scalar > &lla, const Scalar &bearing, const Scalar &distance, const EarthModel &m=EarthModel::WGS84())
template<typename Scalar>
Scalar	horizon_distance (const Scalar &altitude, const EarthModel &m=EarthModel::WGS84())
template<typename Scalar>
Scalar	is_visible (const LLA< Scalar > &lla_observer, const LLA< Scalar > &lla_target, const EarthModel &m=EarthModel::WGS84())
template<typename Scalar>
RayIntersection< Scalar >	ray_ellipsoid_intersection (const Vec3< Scalar > &origin, const Vec3< Scalar > &direction, const EarthModel &m=EarthModel::WGS84())

Function Documentation

◆ destination_point()

template<typename Scalar>

LLA< Scalar > vulcan::geodetic::destination_point	(	const LLA< Scalar > &	lla,
		const Scalar &	bearing,
		const Scalar &	distance,
		const EarthModel &	m = EarthModel::WGS84() )

Compute destination point given start, bearing, and distance

Uses the Vincenty direct formula to compute the endpoint of a geodesic given start point, initial bearing, and distance along the ellipsoid.

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

lla	Starting point
bearing	Initial azimuth [rad], clockwise from North
distance	Distance along ellipsoid [m]
m	Earth model (default: WGS84)

Returns: Destination LLA (altitude set to source altitude)

◆ final_bearing()

template<typename Scalar>

Scalar vulcan::geodetic::final_bearing	(	const LLA< Scalar > &	lla1,
		const LLA< Scalar > &	lla2,
		const EarthModel &	m = EarthModel::WGS84() )

Final bearing arriving at point 2 from point 1

Computes the final bearing (back azimuth + π) arriving at point 2. This is the reverse of initial_bearing(lla2, lla1) + π.

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

lla1	First point (start)
lla2	Second point (destination)
m	Earth model (default: WGS84)

Returns: Azimuth in radians [0, 2π), measured clockwise from North

◆ great_circle_distance()

template<typename Scalar>

Scalar vulcan::geodetic::great_circle_distance	(	const LLA< Scalar > &	lla1,
		const LLA< Scalar > &	lla2,
		const EarthModel &	m = EarthModel::WGS84() )

Great-circle distance using Vincenty formula (accurate to 0.5mm)

Uses Vincenty's inverse formula for geodesic distance on an ellipsoid. This is an iterative algorithm that converges for most point pairs.

Reference: Vincenty, T. (1975). "Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations"

Note: For symbolic mode, uses a fixed iteration count (20 iterations) rather than convergence checking.

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

lla1	First point (altitude ignored)
lla2	Second point (altitude ignored)
m	Earth model (default: WGS84)

Returns: Distance along ellipsoid surface [m]

◆ haversine_distance()

template<typename Scalar>

Scalar vulcan::geodetic::haversine_distance	(	const LLA< Scalar > &	lla1,
		const LLA< Scalar > &	lla2,
		double	radius = constants::earth::R_mean )

Haversine distance (fast spherical approximation)

Uses the haversine formula for great-circle distance on a sphere. Good for short distances (<100km), error <0.3% compared to Vincenty.

Formula: a = sin²(Δφ/2) + cos(φ₁)cos(φ₂)sin²(Δλ/2) c = 2·atan2(√a, √(1−a)) d = R·c

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

lla1	First point (altitude ignored)
lla2	Second point (altitude ignored)
radius	Sphere radius [m] (default: WGS84 mean radius)

Returns: Distance along sphere surface [m]

◆ horizon_distance()

template<typename Scalar>

Scalar vulcan::geodetic::horizon_distance	(	const Scalar &	altitude,
		const EarthModel &	m = EarthModel::WGS84() )

Distance to geometric horizon from given altitude

Computes the distance along the Earth's surface from a point at altitude h to the geometric horizon (line-of-sight tangent to surface).

For a sphere: d = √(2Rh + h²) For ellipsoid: uses mean radius at given position.

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

altitude	Height above ellipsoid [m]
m	Earth model (default: WGS84)

Returns: Distance to horizon along surface [m]

◆ initial_bearing()

template<typename Scalar>

Scalar vulcan::geodetic::initial_bearing	(	const LLA< Scalar > &	lla1,
		const LLA< Scalar > &	lla2,
		const EarthModel &	m = EarthModel::WGS84() )

Initial bearing (azimuth) from point 1 to point 2

Computes the initial bearing (forward azimuth) for the geodesic from point 1 to point 2. This is the direction to head from point 1.

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

lla1	First point (start)
lla2	Second point (destination)
m	Earth model (default: WGS84) – used only for flattening

Returns: Azimuth in radians [0, 2π), measured clockwise from North

◆ is_visible()

template<typename Scalar>

Scalar vulcan::geodetic::is_visible	(	const LLA< Scalar > &	lla_observer,
		const LLA< Scalar > &	lla_target,
		const EarthModel &	m = EarthModel::WGS84() )

Check if target is visible from observer (no terrain, pure geometry)

Uses geometric horizon distance and great-circle arc comparison. Returns a value > 0 if visible, ≤ 0 if not visible.

The visibility check accounts for both observer and target altitudes: visible if horizon_distance(alt1) + horizon_distance(alt2) ≥ ground_distance

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

lla_observer	Observer position
lla_target	Target position
m	Earth model (default: WGS84)

Returns: Positive if visible, zero/negative if occluded by Earth

◆ ray_ellipsoid_intersection()

template<typename Scalar>

RayIntersection< Scalar > vulcan::geodetic::ray_ellipsoid_intersection	(	const Vec3< Scalar > &	origin,
		const Vec3< Scalar > &	direction,
		const EarthModel &	m = EarthModel::WGS84() )

Ray-ellipsoid intersection for terrain/visibility calculations

Computes intersection of ray with WGS84 ellipsoid. Ray: P(t) = origin + t * direction

Ellipsoid equation: (x/a)² + (y/a)² + (z/b)² = 1 Substituting ray equation gives quadratic: At² + Bt + C = 0

Template Parameters

Scalar Scalar type (double for numeric, SymbolicScalar for symbolic)

Parameters

origin	Ray origin in ECEF [m]
direction	Ray direction (will be normalized internally)
m	Earth model (default: WGS84)

Returns: RayIntersection with intersection parameters and points

Classes

Functions

Function Documentation

◆ destination_point()

◆ final_bearing()

◆ great_circle_distance()

◆ haversine_distance()

◆ horizon_distance()

◆ initial_bearing()

◆ is_visible()

◆ ray_ellipsoid_intersection()