Ellipse thing

Expression 12: "l" Subscript, "h" "l" , Baseline equals left brace, alpha equals 0 : "a" , alpha equals 90 : "b" , alpha equals 180 : "a" , alpha equals negative 180 : "a" , alpha equals negative 90 : "b" , StartRoot, left parenthesis, StartFraction, "a" StartRoot, StartNestedFraction, "a" squared tan squared left parenthesis, "A" Subscript, "h" "l" , Baseline , right parenthesis NestedOver "b" squared , EndNestedFraction plus 1 , EndRoot Over StartNestedFraction, "a" squared tan squared left parenthesis, "A" Subscript, "h" "l" , Baseline , right parenthesis NestedOver "b" squared , EndNestedFraction plus 1 , EndFraction , right parenthesis squared plus left parenthesis, StartFraction, "a" StartRoot, StartNestedFraction, "a" squared NestedOver "b" squared , EndNestedFraction plus cot squared left parenthesis, "A" Subscript, "h" "l" , Baseline , right parenthesis , EndRoot Over StartNestedFraction, "a" squared NestedOver "b" squared , EndNestedFraction plus cot squared left parenthesis, "A" Subscript, "h" "l" , Baseline , right parenthesis , EndFraction , right parenthesis squared , EndRoot , right brace

equals

3.9 6 4 5 1 4 1 7 1 1

Expression 13: "l" Subscript, "v" "l" , Baseline equals left brace, alpha equals 0 : "b" , alpha equals 90 : "a" , alpha equals 180 : "b" , alpha equals negative 180 : "b" , alpha equals negative 90 : "a" , StartRoot, left parenthesis, StartFraction, "a" StartRoot, StartNestedFraction, "a" squared tan squared left parenthesis, "A" Subscript, "v" "l" , Baseline , right parenthesis NestedOver "b" squared , EndNestedFraction plus 1 , EndRoot Over StartNestedFraction, "a" squared tan squared left parenthesis, "A" Subscript, "v" "l" , Baseline , right parenthesis NestedOver "b" squared , EndNestedFraction plus 1 , EndFraction , right parenthesis squared plus left parenthesis, StartFraction, "a" StartRoot, StartNestedFraction, "a" squared NestedOver "b" squared , EndNestedFraction plus cot squared left parenthesis, "A" Subscript, "v" "l" , Baseline , right parenthesis , EndRoot Over StartNestedFraction, "a" squared NestedOver "b" squared , EndNestedFraction plus cot squared left parenthesis, "A" Subscript, "v" "l" , Baseline , right parenthesis , EndFraction , right parenthesis squared , EndRoot , right brace

equals

3.6 6 9 4 0 8 8 7 5 6 5

Hidden Label: "H" equals left parenthesis, left bracket, 1 , negative 1 , right bracket left brace, 0 less than cos left parenthesis, alpha , right parenthesis : 1 , cos left parenthesis, alpha , right parenthesis less than 0 : negative 1 , right brace "l" Subscript, "h" "l" , Baseline cos left parenthesis, alpha plus "A" Subscript, "h" "l" , Baseline , right parenthesis , left bracket, 1 , negative 1 , right bracket left brace, 0 less than cos left parenthesis, alpha , right parenthesis : 1 , cos left parenthesis, alpha , right parenthesis less than 0 : negative 1 , right brace "l" Subscript, "h" "l" , Baseline sin left parenthesis, alpha plus "A" Subscript, "h" "l" , Baseline , right parenthesis , right parenthesis

Label

equals

left parenthesis, 3.5 3 2 , negative 1.8 0 1 , right parenthesis

left parenthesis, negative 3.5 3 2 , 1.8 0 1 , right parenthesis

Expression 15: "x" equals "H" . "x"

equals

3.5 3 1 9 1 0 8 8 2 4 8

negative 3.5 3 1 9 1 0 8 8 2 4 8

Hidden Label: "V" equals left parenthesis, left bracket, 1 , negative 1 , right bracket left brace, 0 less than sin left parenthesis, alpha , right parenthesis : 1 , sin left parenthesis, alpha , right parenthesis less than 0 : negative 1 , right brace "l" Subscript, "v" "l" , Baseline cos left parenthesis, alpha plus "A" Subscript, "v" "l" , Baseline , right parenthesis , left bracket, 1 , negative 1 , right bracket left brace, 0 less than sin left parenthesis, alpha , right parenthesis : 1 , sin left parenthesis, alpha , right parenthesis less than 0 : negative 1 , right brace "l" Subscript, "v" "l" , Baseline sin left parenthesis, alpha plus "A" Subscript, "v" "l" , Baseline , right parenthesis , right parenthesis

Label

equals

left parenthesis, negative 2.9 9 , 2.1 2 7 , right parenthesis

left parenthesis, 2.9 9 , negative 2.1 2 7 , right parenthesis

Expression 17: "y" equals "V" . "y"

equals

2.1 2 7 3 4 7 0 6 1 0 7

negative 2.1 2 7 3 4 7 0 6 1 0 7

Epiphany

Expression 32: "l" Subscript, "A" , Baseline equals left bracket, 0 , 7.5 , 15 , 22.5 , 30 , 37.5 , 45 , 52.5 , 60 , 67.5 , 75 , 82.5 , 90 , 97.5 , 105 , 112.5 , 120 , 127.5 , 135 , 142.5 , 150 , 157.5 , 165 , 172.5 , 180 , right bracket

25 element list

The following equation shows how the angles above can be used to show how they would appear on the ellipse (you can sort of make out the ellipse's shape toward where the lines converge). If you subtract the resulting values from their respective angle in the list, you get the angle needed to calculate the length from the center of the ellipse to its horizontal max and min.

Expression 34: "y" cosine left parenthesis, tan to the negative 1st power left parenthesis, StartFraction, "b" squared sin left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis Over "a" squared cos left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis , EndFraction , right parenthesis , right parenthesis equals "x" sine left parenthesis, tan to the negative 1st power left parenthesis, StartFraction, "b" squared sin left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis Over "a" squared cos left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis , EndFraction , right parenthesis , right parenthesis

These angles are used to find the vertical max and min

Expression 36: "y" cosine left parenthesis, tan to the negative 1st power left parenthesis, StartFraction, "b" squared cos left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis Over "a" squared sin left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis , EndFraction , right parenthesis , right parenthesis equals "x" sine left parenthesis, tan to the negative 1st power left parenthesis, StartFraction, "b" squared cos left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis Over "a" squared sin left parenthesis, "l" Subscript, "A" , Baseline , right parenthesis , EndFraction , right parenthesis , right parenthesis