APPROXIMATING AN ELLIPSE WITH FOUR CIRCULAR ARCS
Tirupathi R. Chandrupatla
Mechanical Engineering Department
Thomas J. Osler
Mathematics Department
Rowan University
Glassboro, New Jersey 08028
chandrupatla@rowan.edu
osler@rowan.edu
1. Introduction
Ellipses find wide applications in mathematics, physics, and engineering. Elliptic
shapes are used in architectural and design forms. Every circular shape turns into an
ellipse when viewed at an angle and is thus an elementary geometric form in drafting.
Elliptic forms are difficult to produce since they have a continuously varying curvature. If
an elliptic arch is to be built, each piece of the structure needs to be of a different shape.
In addition, a parallel curve to an ellipse is not an ellipse [1]. This finds important
application in engineering. This is used for generating tool paths in computer aided
machining [2]. Curves using piecewise circular arcs overcome these difficulties. There
are several interesting approaches to approximating ellipses using circular arcs. A
detailed discussion of the historical and mathematical aspects of these approximations
has been presented in a series of papers by [5]. The most interesting and widely used
constructions use four circular arcs (also referred to as quadrarc) to approximate an
1
ellipse. The simplicity of construction using a compass and a ruler also results in
simplified computations using a computer. In section 2 we present a clear geometric
derivation of the mathematical theory that governs all four arc constructions of the ellipse
x 2 / a 2 + y 2 / b 2 = 1 . In section 3 we summarize the construction method and in section 4
we describe the most frequently used construction known as French’s construction.
This material uses only precalculus mathematic. It is particularly suitable for
courses in geometry. Engineering majors might find it especially interesting.
2. The geometry of all four arc approximations to an ellipse
We make use of the symmetry of the ellipse about its major and minor axes. Thus the
four arc construction is reduced to the two arc configuration shown in Fig.1. The basic
requirements of the four circular arc approximation of the ellipse are set as follows:
a) the arcs pass through the extremal points A and B of the ellipse,
b) the arcs have tangent continuity at T where they join .
Arc 1 is tangent to side AE and passes through the extreme point A, and arc 2 is tangent to
side BE and passes through the extreme point B. The arcs meet at the transition point T,
where they have a common tangent. These conditions lead to the following
requirements:
2
a
B
E
*
T
*
r2
O
C1
b
×
θ
×
A
r1
C2
Figure 1. Two-arc configuration
• center C1 of arc l of radius r1 lies on line AO,
• center C2 of arc 2 of radius r2 lies on line BO,
• centers C1 and C2 lie on the common normal through the transition point T. We
refer to this normal as the transition normal.
Triangles ATC1 and BTC2 in Fig.1 are isosceles triangles with equal sides of r1 and r2
respectively. We denote angle AC1T as θ .
3
B
(0, b)
E
Q
π /4
T
π / 4 +θ / 2
π / 4 −θ / 2
O
(0, 0)
×
×
×
×
C1
C
φ
R
A
(a, 0)
× =θ /2
* = π / 4 −θ / 2
* *
a −b
2
C2
Figure 2. Arc transition condition
The following is a concise outline of the steps needed to derive our geometric vision
of all four arc constructions. We invite the reader to fill in the details.
1.
Fig. 2 is a modification of Fig. 1 with additional construction lines. The
perpendicular drawn from C1 to AT bisects angle AC1T and the perpendicular drawn from
C2 to BT bisects angle BC2T . These two perpendiculars intersect at point C. C is the
center for the arc ATB. Since CC1 bisects angle OC1C2 , and CC2 bisects angle OC2C1 ,
we note that C is also the incenter of triangle OC1C2 .
4
2.
From the angle values marked in the figure, we find after a little calculation that the
external angle BTQ of triangle ATB is
BTQ =
π
4
.
(1)
This is true for every transition point T.
3.
From the geometry of a circle, we note that if C is its center, and A, T and B are
three consecutive points on its boundary, then the central angle ACB is twice the external
angle BTQ. Since angle BTQ = π / 4 from Equation 1, angle ACB is π / 2 . We can also
see this by noting that angle OAC = OBC and angle AOB is π / 2 . Thus the transition
point T lies on an arc of a circle with chord AB subtending an angle of π / 2 at its center
C. This arc is the trajectory of all transition points and we call it the transition arc.
4.
The center C is located as shown in Fig. 3. A semicircle is
B
(0, b)
E
d/2
a/2
M
d = AB
b/2
d/2
b/2
a/2
O
(0, 0)
R
φ
A
(a, 0)
C
⎛ a −b a −b⎞
,−
⎜
⎟
2 ⎠
⎝ 2
Figure 3. Center of the transition arc
5
drawn with AB as diameter. The perpendicular bisector MC of AB intersects the
semicircle at C, which is the center of the transition arc. Four identical right triangles of
sides a/2, b/2, and d/2 ( d = AB ) are drawn to show that the center C has coordinates
⎛ a −b a −b ⎞
,−
⎜
⎟.
2 ⎠
⎝ 2
5.
The radius R of this arc is d / 2 which is
R=
6.
The inradius of triangle OC1C2 is
a2 + b2
2
(2)
a −b
. Thus the transition normal C2C1T is
2
tangent to the incircle of the triangle OC1C2 , which has a fixed radius.
The above tangent continuity conditions may be summarized as follows.
Tangent continuity conditions:
1) The transition point must lie on the transition arc with its center C at
a 2 + b2
⎛ a −b a −b ⎞
,
−
R
,
and
radius
=
.
⎜
⎟
2 ⎠
2
⎝ 2
2) The transition normal must be tangential to the circle of radius
a −b
, with its
2
⎛ a −b a −b ⎞
,−
center C at ⎜
⎟ . We call this the constraint circle.
2 ⎠
⎝ 2
Thus these two concentric circles play an important role in the ellipse approximation.
The above two conditions are given in [5], where the second condition is given as an
algebraic constraint. We now add the angle ω in Fig. 4 and establish the relationships to
6
calculate the arc radii. Angles θ and ω have a constant difference of π / 2 − 2φ , where
φ is angle OAB.
⎛ω ⎞
R sin ⎜ ⎟
⎝2⎠
r1 =
ω⎞
⎛π
sin ⎜ − φ + ⎟
2⎠
⎝4
(3)
⎛π ω ⎞
R sin ⎜ − ⎟
⎝4 2⎠
r2 =
ω⎞
⎛
sin ⎜ φ − ⎟
2⎠
⎝
where R =
a 2 + b2
, and ω = θ + 2φ − π / 2 .
2
D
B
(0, b)
E
Q
π /4
π /4
π / 4 −φ
T
R
π / 2 − 2φ
θ
O
(0, 0)
ω
φ
A
(a, 0)
C1
π / 4 −φ
π/4−φ
C
π / 2 −θ
r1 = C1T = C1 A
Angle ACD = 2φ
r2 = C2T = C2 B θ = π / 2 − 2φ + ω
R=
C2
Figure 4. Arc parameters
7
a2 + b2
2
From Fig. 4, we see that r1 + ( r2 − r1 ) cos θ = a and r2 − ( r2 − r1 ) sin θ = b . From these,
we get alternative expressions for the radii
a sin θ + b cos θ − a
sin θ + cos θ − 1
a sin θ + b cos θ − b
r2 =
sin θ + cos θ − 1
r1 =
(4)
We observe from (3) that when ω is close to zero r1 tends to zero. At ω = 2φ , the
transition point T is at D, and the radius r2 tends to infinity. This corresponds to arc 2
being the straight line BD. For angles in the range 2φ < ω < π / 2 , radius r2 becomes
negative. In this range, point T is outside the rectangle enclosed by a, b. Thus the arcs of
interest are formed when the transition point T is strictly in the interval AD of the
transition arc.
3. Summary of construction methods
A general construction may be described as follows.
a. Draw the rectangle OAEB (Fig.4) with sides a and b.
a −b
⎛ a −b a −b ⎞
,−
b. Locate the center C at ⎜
and a
⎟ and draw the incircle with radius
2 ⎠
2
⎝ 2
concentric arc ADB with radius CA.
c. Choose a transition point T strictly in the interval AD of the transition arc. From the
transition point draw the tangent line to the constraint circle. This line is the transition
normal which intersects OA and OB at the arc centers C1 and C2 respectively.
8
d. Arcs AT and BT are drawn using these centers C1 and C2 .
We now present the most commonly used construction of this type.
4. French’s Construction
The construction shown in Fig. 5 is referred to as French’s construction [3]. This
construction is one of the most widely used ellipse approximations in geometric drawing.
a
B
E
a–b
T
L
b
K
r2
φ
θ
O
θ
A
r1
C1
φ
C2
Figure 5. French’s Four-Arc Approximation of Ellipse
Given the half major and minor axes as a and b, BL is set equal to a − b. KC1C2 is drawn
as the perpendicular bisector of AL. Then using C1 A as radius, arc AT is drawn, and using
C2 B as radius, arc BT is drawn. Curve ATB is the approximation to the ellipse with semimajor and semi-minor axes a and b. Two other centers are placed in a symmetrically and
9
the full ellipse is drawn. We now proceed to discuss how this construction fits into the
four-arc approximation conditions presented in section 2. Consider the case when the
transition normal is perpendicular to diagonal AB as shown in Fig.6.
Denoting θ = angle AC1 K , and φ = angle BC2 K , where θ + φ = π / 2 , we have angle
ATK = π / 2 − θ / 2 in the isosceles triangle AC1T , and angle BTK = π / 2 − φ / 2 in the
isosceles triangle BC2T . From the right triangles ATK and BTK, we have angle
TAK = θ / 2 and angle TBK = φ / 2 . Thus point T is the intersection of angle bisectors of
triangle
ABE, and it is the incenter of the triangle. A property of the incenter is that BK = s − b
and AK = s − a , where s = ( AB + a + b ) / 2 , see [4]. We note that
BK − AK = a − b . Setting
a
B
E
a – b φ /2
L
T
b
K
θ /2
θ
O
φ
A
C1
φ
C2
Figure 6. Transition Normal Perpendicular to AB
10
point L at a distance a − b from B, K is the midpoint of AL. Line C2C1 KT is the
perpendicular bisector of AL. The basic idea of French’s construction is now clear. The
reader may also verify that the incenter is the midpoint of arc AD in Fig. 4.
5. How good is our construction?
The circular arc 1 passes through A and a major portion of this arc is above the ellipse.
We denote this largest positive error by ε1 . The circular arc 2 passes through B and a
major part of this arc is below the ellipse. This negative error of largest magnitude is
denoted as ε 2 . We show the maximum errors ε max = max ( ε1 , −ε 2 ) in Table 1 for six
different ellipses. These are calculated using formulas from Qian and Qian [6] that are
beyond the scope of this paper.
Table 1
a
b
r1
r2
θ
Max error ε max
1.1
1.2
1.5
2
4
6
1
1
1
1
1
1
0.9370
0.8865
0.7829
0.6910
0.5788
0.5488
1.1793
1.3762
2.0757
3.6180
14.6847
33.7069
0.8330
0.8761
0.9828
1.1071
1.3258
1.4056
0.0024
0.0052
0.0157
0.0361
0.1101
0.1603
We note that the error increases when the aspect ratio a/b increases.
11
References
[1] Banchoff, T. and Giblin, P. (1994) On the geometry of piecewise circular curves
Amer. Math. Monthly 101, 403-416. ISSN: 0002-9890
[2] Elber, G., Lee, I.K. and Kim, M.S. (1997) Comparing offset curve approximation
methods, IEEE Computer Graphics and Applications 17, 62-71. ISSN: 0272-1716
[3] French, T.E. (1947) A manual of engineering drawing (7th Ed), Mc Graw Hill, New
York, 1947. ISBN: 0795919752
[4] Osler, T.J. and Fine, I. (2001) The remarkable incircle of a triangle, Mathematics and
Computer Education 35, 44-50. ISSN: 0730 - 8639
[5] Rosin, P.L. (1999) A survey and comparison of traditional piecewise circular
approximations to the ellipse, Computer Aided Geometric Design 16, 269-286. ISSN:
0167-8396
[6] Qian, W-H. and Qian, K, (2001) Optimising the four-arc approximation to ellipses,
Computer Aided Geometric Design 18, 1-19. ISSN: 0167-8396
12