Machinery's Handbook, 31st Edition
CIRCLE FORMULAS
51
E 2 ⁄ R F – ------- 13.27 2 ⁄ 12– 2 =
φ 2 -- tan
φ 2 --
---------- 0.6635, =
=
=
33.56 °,
67.13 ° =
φ
tan –1 0.6635 =
π
π
L R φ = × = 12 × 67.13° ×
= 14.06 cm
180
180
---- E R F – ( ) 2
– ----------- 12 14.06 ( ) 2 =
------------ 13.27 10 ( ) 2
– ------------ 84.36 – 66.35 18.01 cm 2 = =
Area S RL 2 =
Another way to find angle f is divide the chord length by twice the radius to obtain φ 2 -- sin chord length 2 R --------------- E 2 R ---- 0.5529, = = = φ 2 -- 33.5662 °, = φ sin –1 0.5529 = Ellipse.— As a circle is the locus of points equidistant from a single point in the plane, an ellipse is the set of points whose location is established by two points in the plane. Refer- ring to the figure, these two points are the foci , F 1 and F 2 , which lie on the longer of the two diameters of the ellipse. The longer diameter a is the major axis; the shorter b is the minor axis. The sum of the distances from the foci to any point P on the ellipse is constant. That is PF 1 + PF 2 = 2 a . The latus rectum is the chord through the focus and perpendicular to the major axis. V 1 and V 2 , are the vertices. Like the circle, there is a general form and a standard form of the equation of an ellipse. The general form is: Ax 2 Cy 2 Dx Ey F + + + + = 0 AC 0 and A C ≠ > The constant F in this equation is not related to the foci, which are not numbers, but labels for points. 67.13 ° =
Ellipse If ( h , k ) is the center, the standard equation of an ellipse is x h – ( ) 2 a 2 ----------
2
y k – ( ) 2 + --------- = 1 .
b
The eccentricity e of the ellipse is given by e = c / a , where c 2 = a 2 – b 2 . This is not the same e as the exponential base, which is a constant. Rather, eccentricity varies with the figure. e a 2 b 2 – a ----------- c a -- = = is a measure of the elongation of the ellipse and is al- . The aspect ratio of the ellipse is a / b . The equation of an ellipse centered at ( h , k ) = (0, 0) with foci at ( ± c , 0) is x 2 a 2 --- y 2 b 2 + --- = 1 , and the ellipse is symmetric about both coordinate axes. Its ways less than 1. The distance between the two foci is 2 c 2 a 2 b 2 – = x -intercepts are ( ± a , 0) and y -intercepts are (0, ± b ). The line segment joining (0, b ) and (0, - b ) is called the minor axis. The major vertices of the ellipse are ( ± a , 0), and the line segment joining vertices V 1 and V 2 is the major axis of the ellipse.
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