Statements (83)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
Hypercomplex number |
| gptkbp:algebraicStructure |
Non-commutative division algebra
|
| gptkbp:alternativeName |
Quadric
|
| gptkbp:basisFor |
1
i k j |
| gptkbp:category |
gptkb:geometry
gptkb:Linear_algebra gptkb:Abstract_algebra Mathematical physics |
| gptkbp:class |
By rank of matrix
By signature of matrix |
| gptkbp:component |
gptkb:Real_number
|
| gptkbp:definedIn |
A hypersurface in n-dimensional space defined by a second-degree polynomial equation
|
| gptkbp:degree |
2
|
| gptkbp:dimensions |
4
n-dimensional |
| gptkbp:example |
gptkb:Circle
gptkb:Parabola gptkb:Sphere gptkb:Ellipsoid gptkb:Hyperbola gptkb:Hyperbolic_paraboloid Ellipse Hyperboloid Paraboloid |
| gptkbp:field |
gptkb:Mathematics
gptkb:Algebraic_geometry Real numbers |
| gptkbp:generalizes |
Complex number
|
| gptkbp:hasApplication |
gptkb:Molecular_dynamics
gptkb:signal_processing gptkb:Animation gptkb:Quantum_mechanics Computer vision Aerospace engineering Virtual reality Attitude control Robotics kinematics |
| gptkbp:hasConjugate |
Quaternion conjugate
|
| gptkbp:hasEquation |
Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
|
| gptkbp:hasInverse |
Quaternion inverse
|
| gptkbp:hasMatrixRepresentation |
Symmetric matrix
|
| gptkbp:hasNorm |
Quaternion norm
|
| gptkbp:hasSpecialCase |
Conic section
|
| gptkbp:introducedIn |
1843
|
| gptkbp:inventedBy |
gptkb:William_Rowan_Hamilton
|
| gptkbp:multiplicationRule |
i^2 = j^2 = k^2 = ijk = -1
ij = k ik = -j ji = -k jk = i ki = j kj = -i |
| gptkbp:notation |
q = a + bi + cj + dk
|
| gptkbp:property |
Can be classified by eigenvalues of associated matrix
Can be degenerate or non-degenerate Symmetric with respect to its principal axes |
| gptkbp:relatedGroup |
Unit quaternions form SU(2)
|
| gptkbp:relatedTo |
Complex number
Quadratic form Matrix representation Second-degree polynomial Octonion Rotation group SO(3) |
| gptkbp:studiedIn |
gptkb:Linear_algebra
gptkb:Analytic_geometry gptkb:Projective_geometry |
| gptkbp:usedFor |
Interpolating orientations
Representing rotations |
| gptkbp:usedIn |
gptkb:Computer_graphics
gptkb:Physics gptkb:robot gptkb:Control_theory gptkb:Theoretical_physics Engineering Navigation 3D computer graphics Geometric modeling |
| gptkbp:bfsParent |
gptkb:UnityEngine
|
| gptkbp:bfsLayer |
5
|