Editing Hyperboloid (section)

==Relation to the sphere==
In 1853 [[William Rowan Hamilton]] published his ''Lectures on Quaternions'' which included presentation of [[biquaternion]]s.  The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from [[quaternion]]s to produce hyperboloids from the equation of a [[sphere]]:
<blockquote>... the ''equation of the unit sphere'' {{math|1=''ρ''<sup>2</sup> + 1 = 0}}, and change the vector {{math|''ρ''}} to a ''bivector form'', such as {{math|''σ'' + ''τ'' {{radic|−1}}}}.  The equation of the sphere then breaks up into the system of the two following,
{{block indent | em = 1.5 | text = {{math|1=''σ''<sup>2</sup> &minus; ''τ''<sup>2</sup> + 1 = 0}}, {{math|1='''S'''.''στ'' = 0}};}}
and suggests our considering {{math|''σ''}} and {{math|''τ''}} as two real and rectangular vectors, such that
{{block indent | em = 1.5 | text = {{math|1='''T'''''τ'' = ('''T'''''σ''<sup>2</sup> &minus; 1 )<sup>1/2</sup>}}.}}
Hence it is easy to infer that if we assume {{math|''σ'' {{!!}} ''λ''}}, where {{math|''λ''}} is a vector in a given position, the ''new real vector'' {{math|''σ'' + ''τ''}} will terminate on the surface of a ''double-sheeted and equilateral hyperboloid''; and that if, on the other hand, we assume {{math|''τ'' {{!!}} ''λ''}}, then the locus of the extremity of the real vector {{math|''σ'' + ''τ''}} will be an ''equilateral but single-sheeted hyperboloid''. The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ...</blockquote>
In this passage {{math|'''S'''}} is the operator giving the scalar part of a quaternion, and {{math|'''T'''}} is the "tensor", now called [[norm (mathematics)|norm]], of a quaternion.

A modern view of the unification of the sphere and hyperboloid uses the idea of a [[conic section]] as a [[conic section#As slice of quadratic form|slice of a quadratic form]]. Instead of a [[conical surface]], one requires conical [[hypersurface]]s in [[four-dimensional space]] with points {{math|1=''p'' = (''w'', ''x'', ''y'', ''z'') &isin; '''R'''<sup>4</sup>}} determined by [[quadratic form]]s. First consider the conical hypersurface
*<math>P = \left\{ p \; : \; w^2 = x^2 + y^2 + z^2 \right\} </math> and
*<math>H_r = \lbrace p \ :\  w = r \rbrace ,</math> which is a [[hyperplane]].
Then <math>P \cap H_r</math> is the sphere with radius {{math|''r''}}. On the other hand, the conical hypersurface
{{block indent | em = 1.5 | text = <math>Q = \lbrace p \ :\  w^2 + z^2 = x^2 + y^2 \rbrace</math> provides that <math>Q \cap H_r</math> is a hyperboloid.}}

In the theory of [[quadratic form]]s, a '''unit [[quasi-sphere]]''' is the subset of a quadratic space {{math|''X''}} consisting of the {{math|''x'' &isin; ''X''}} such that the quadratic norm of {{math|''x''}} is one.<ref>[[Ian R. Porteous]] (1995) ''Clifford Algebras and the Classical Groups'', pages 22, 24 & 106, [[Cambridge University Press]] {{ISBN|0-521-55177-3}}</ref>