Coordinate System#

NavalToolbox uses a right-handed coordinate system consistent with standard naval architecture conventions.

Axes Definition#

Axis

Direction

Sign Convention

X

Longitudinal (length)

Positive towards bow (forward)

Y

Transverse (beam)

Positive towards port (left when facing bow)

Z

Vertical (height)

Positive upward

Note

The origin is typically at the intersection of the aft perpendicular (AP), centerline, and baseline (keel), but may vary depending on the hull geometry file.

Rotations#

Rotations follow the right-hand rule: curl your fingers around the axis in the direction of positive rotation, and your thumb points in the positive axis direction.

Rotation

Axis

Positive Direction

Heel (φ)

Around X-axis

Port UP / Starboard down

Trim (θ)

Around Y-axis

Bow down / Stern up

Physical Interpretation#

Heel:

  • Positive heel (φ > 0): Port side goes UP, starboard side goes DOWN

  • Negative heel (φ < 0): Port side goes DOWN, starboard side goes UP

Trim:

  • Positive trim (θ > 0): Bow goes DOWN, stern goes UP

  • Negative trim (θ < 0): Bow goes UP, stern goes DOWN (stern trim)

Equilibrium Angles#

When calculating equilibrium from a center of gravity (COG) offset:

  • TcG > 0 (weight on port): Results in negative heel (port goes down)

  • TcG < 0 (weight on starboard): Results in positive heel (starboard goes down)

  • LcG > LCB (weight forward of buoyancy): Results in positive trim (bow down)

  • LcG < LCB (weight aft of buoyancy): Results in negative trim (stern down)

GZ Curve Convention#

The righting arm GZ is calculated as:

\[GZ = G_y^{world} - B_y^{world}\]

Where \(G_y^{world}\) and \(B_y^{world}\) are the Y-coordinates of the center of gravity and center of buoyancy in the world frame (waterplane horizontal).

  • Positive GZ: Restoring moment (ship tends to return upright)

  • Negative GZ: Capsizing moment (ship tends to heel further)

At small heels, this simplifies to: \(GZ \approx GM \cdot \sin(\phi)\)