Peanut Surfaces: Trace as "circle of best fit" to <Ax,x>x

Expression 18: "S" left parenthesis, "u" , "v" , right parenthesis equals left parenthesis, "s" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis , "s" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis , "s" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis , right parenthesis

thanks to Desmos demo "parameterized surface of revolution" for teaching how to make surfaces

Expression 20: "S" left parenthesis, "u" , "v" , right parenthesis

domain u Minimum: 0

less than or equal to "u" less than or equal to

domain u Maximum: 2 pi

domain v Minimum: negative pi

less than or equal to "v" less than or equal to

domain v Maximum: pi

Expression 21: "e" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis equals "d" Subscript, 1 , Baseline "s" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis plus "a" "s" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis plus "b" "s" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis

Expression 22: "e" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis equals "a" Subscript, 2 , Baseline "s" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis plus "d" Subscript, 2 , Baseline "s" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis plus "c" "s" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis

Expression 23: "e" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis equals "b" Subscript, 2 , Baseline "s" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis plus "c" Subscript, 2 , Baseline "s" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis plus "d" Subscript, 3 , Baseline "s" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis

Expression 24: "E" left parenthesis, "u" , "v" , right parenthesis equals left parenthesis, "e" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis , "e" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis , "e" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis , right parenthesis

Expression 25: "E" left parenthesis, "u" , "v" , right parenthesis

domain u Minimum: 0

less than or equal to "u" less than or equal to

domain u Maximum: 2 pi

domain v Minimum: negative pi

less than or equal to "v" less than or equal to

domain v Maximum: pi

Expression 26: "S" left parenthesis, alpha , beta , right parenthesis

equals

left parenthesis, negative 0.2 2 6 6 4 5 6 8 , 0.9 7 1 4 6 2 6 8 , 0.0 6 9 9 4 2 8 5 , right parenthesis

Expression 27: "E" left parenthesis, alpha , beta , right parenthesis

equals

left parenthesis, 0.3 5 6 2 3 1 9 3 , 1.8 0 6 9 3 7 9 5 , 0.0 6 9 9 4 2 8 5 , right parenthesis

Expression 28: alpha equals 1.8

2 pi

Expression 29: beta equals 0.0 7

negative pi

Expression 30: "P" left parenthesis, "u" , "v" , right parenthesis equals left parenthesis, "s" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis "g" left parenthesis, "u" , "v" , right parenthesis , "s" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis "g" left parenthesis, "u" , "v" , right parenthesis , "s" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis "g" left parenthesis, "u" , "v" , right parenthesis , right parenthesis

Expression 31: "g" left parenthesis, "u" , "v" , right parenthesis equals "f" left parenthesis, "s" Subscript, 1 , Baseline left parenthesis, "u" , "v" , right parenthesis , "s" Subscript, 2 , Baseline left parenthesis, "u" , "v" , right parenthesis , "s" Subscript, 3 , Baseline left parenthesis, "u" , "v" , right parenthesis , right parenthesis

Expression 32: "P" left parenthesis, "u" , "v" , right parenthesis

domain u Minimum: 0

less than or equal to "u" less than or equal to

domain u Maximum: 2 pi

domain v Minimum: negative pi

less than or equal to "v" less than or equal to

domain v Maximum: pi

Expression 33: "P" left parenthesis, alpha , beta , right parenthesis

equals

left parenthesis, negative 0.3 8 0 6 5 7 3 9 , 1.6 3 1 5 9 7 1 7 , 0.1 1 7 4 7 0 8 6 , right parenthesis

Expression 34: segment left parenthesis, left parenthesis, 0 , 0 , 0 , right parenthesis , 2 "P" left parenthesis, alpha , beta , right parenthesis , right parenthesis

Expression 35: segment left parenthesis, "P" left parenthesis, alpha , beta , right parenthesis , "E" left parenthesis, alpha , beta , right parenthesis , right parenthesis

Expression 36: "x" squared plus "y" squared plus "z" squared equals left parenthesis, 1 third left parenthesis, "d" Subscript, 1 , Baseline plus "d" Subscript, 2 , Baseline plus "d" Subscript, 3 , Baseline , right parenthesis , right parenthesis squared