ELLIPSE CALCULATIONS Machinery's Handbook, 31st Edition
53
Length, Point, and Angle Calculations R 1 = radius of director circle = A 2 B 2 + , R 2 = radius of equivalent circle = AB , P = center to focus distance = A 2 B 2 – ,
A = major radius = B 2 B = minor radius = A 2
2
+
P
2
–
P
2
2 B
A ----- = distance, origin to latus rectum
A φ = cos , and Y = B
J = any point ( X , Y ) on curve where X = A θ sin
θ cos
B φ = sin
–1 Y B --
–1 X A --
f = angle with major axis = sin
, q = angle with minor axis = 90 ° φ –
=
cos
L = total perimeter (approximate) = A 1.2 B A -- 2
B A -- 4 +
+
1.1
L = perimeter (sections) = π 180 -----
2 φ AB
Area Calculations
N = total surface area of ellipse = π AB W = area between outer and inner ellipse = π A 1 B 1 A 2 B 2 – ( ) M = area of complement section M = AB π AB 4 – ------ S = area of segment S = AB cos –1 X 1 A --- X the inverse cosine is in radian measure T+S = combined area of segment S + area T = AB cos –1 X 2 A --- X that results from the inverse cosine is in radian measure V = area of segment V = R 2 2 sin –1 X A -- XY – ( )
1 Y 1 – , where the angle that results from
2 Y 2 – , where the angle
, where the angle that results from the
inverse sine is in radian measure K = area of sector K = AB cos –1 X A --
, where the angle that results from the inverse
cosine is in radian measure Example 2: Find area of sector K and complement area M , given the major radius of ellipse is 10 cm, minor radius of ellipse is 7 cm, dimension X = 8.2266 cm. Solution: Sectional area K: Area K AB cos –1 X 1 A --- 10 7 cos –1 8.2266 10 -------- × × 70 0.6047 rad × 42.33 cm 2 = = = = = Example 3: Find the area of elliptical segments S, T + S , provided that major radius A of ellipse is 10 cm, minor radius B of ellipse is 7 cm, dimension X 1 = 8.2266 cm, dimension Y 1 = 4.4717 cm, and dimension X 2 = 6.0041 cm. Solution: Segment area S is found: S AB cos –1 X 1 A --- X 1 Y 1 – 10 7 cos –1 8.2266 10 -------- × × 8.2266 4.4717 × – 5.5437 cm 2 = = = Solution: Complement area M: Area M AB π AB 4 = – ------ 10 7 π = 10 7 × × 4 – ------------ × 15.0221 cm 2
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