Expression 6: left parenthesis, StartFraction, "r" Subscript, "a" , Baseline cos left parenthesis, "u" , right parenthesis cos left parenthesis, "v" , right parenthesis Over StartRoot, 1 minus "s" cos left parenthesis, "u" , right parenthesis squared sin left parenthesis, "v" , right parenthesis squared minus "s" sin left parenthesis, "u" , right parenthesis squared , EndRoot , EndFraction , StartFraction, "r" Subscript, "a" , Baseline cos left parenthesis, "u" , right parenthesis sin left parenthesis, "v" , right parenthesis Over StartRoot, 1 minus "s" cos left parenthesis, "u" , right parenthesis squared cos left parenthesis, "v" , right parenthesis squared minus "s" sin left parenthesis, "u" , right parenthesis squared , EndRoot , EndFraction , StartFraction, "r" Subscript, "a" , Baseline sin left parenthesis, "u" , right parenthesis Over StartRoot, 1 minus "s" cos left parenthesis, "u" , right parenthesis squared , EndRoot , EndFraction , right parenthesisracosucosv1−scosu2sinv2−ssinu2,racosusinv1−scosu2cosv2−ssinu2,rasinu1−scosu2
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domain u Minimum:
less than or equal to "u" less than or equal to≤u≤
domain u Maximum: 2 pi2π
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domain v Minimum:
less than or equal to "v" less than or equal to≤v≤
domain v Maximum: piπ
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Expression 7:
7
Implicit version. Has problems when s is close to 1.
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Expression 9: "x" squared plus "y" squared plus "z" squared minus StartFraction, "s" squared Over "r" squared , EndFraction left parenthesis, "x" squared "y" squared plus "y" squared "z" squared plus "z" squared "x" squared , right parenthesis plus StartFraction, "s" to the 4th power Over "r" to the 4th power , EndFraction "x" squared "y" squared "z" squared equals "r" squared left brace, negative "r" Subscript, "a" , Baseline less than or equal to "x" less than or equal to "r" Subscript, "a" , Baseline , right brace left brace, negative "r" Subscript, "a" , Baseline less than or equal to "y" less than or equal to "r" Subscript, "a" , Baseline , right brace left brace, negative "r" Subscript, "a" , Baseline less than or equal to "z" less than or equal to "r" Subscript, "a" , Baseline , right bracex2+y2+z2−s2r2ax2y2+y2z2+z2x2+s4r4ax2y2z2=r2a−ra≤x≤ra−ra≤y≤ra−ra≤z≤ra
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Expression 10: "r" Subscript, "a" , Baseline equals 1ra=1
