Conic Sections
Applied to Aircraft
Dr. S.M. Malaek
Assistant: M. Younesi
Conic Sections
Conic Sections
The Ellipse
„
Though not so simple as the circle, the
ellipse is nevertheless the curve most often
"seen" in everyday life. The reason is that
every circle, viewed obliquely, appears
elliptical.
The Ellipse
„
The early Greek astronomers thought that the
planets moved in circular orbits about an
unmoving earth, since the circle is the simplest
mathematical curve. In the 17th century,
Johannes Kepler eventually discovered that each
planet travels around the sun in an elliptical
orbit with the sun at one of its foci.
The Ellipse
„
The orbits of the moon and of artificial satellites
of the earth are also elliptical as are the paths of
comets in permanent orbit around the sun.
The Ellipse
„
On a far smaller scale, the electrons of an atom
move in an approximately elliptical orbit with
the nucleus at one focus.
The Parabola
„
One of nature's best known approximations to
parabolas is the path taken by a body projected upward
and obliquely to the pull of gravity, as in the parabolic
trajectory of a golf ball. The friction of air and the pull
of gravity will change slightly the projectile's path from
that of a true parabola, but in many cases the error is
The Parabola
„
The easiest way to visualize the path of a
projectile is to observe a waterspout. Each
molecule of water follows the same path and,
therefore, reveals a picture of the curve.
The Parabola
„
Heat waves, as well as light and sound waves,
are reflected to the focal point of a parabolic
surface.
The Hyperbola
„
If a right circular cone is intersected by a
plane parallel to its axis, part of a hyperbola
is formed. Such an intersection can occur in
physical situations as simple as sharpening a
pencil that has a polygonal cross section or
in the patterns formed on a wall by a lamp
shade.
Conic Sections Applied to Aircraft
„
An Aircraft Fuselage
General Graphical
Construction
Technique
General Graphical Construction Technique
„
Curve equation:
B
C
E
D
F
F ( x, y ) = A( x + xy + y + x + y + ) = 0
A
A
A
A
A
2
2
General Graphical Construction Technique
„
„
„
The basic principle of the usual method of constructing
the general conic section :
The tangents to the curve at the points of contact o and
B are AO and AB respectively.
The given control point (fifth condition) through which
it is desired to pass the curve ids D.
General Graphical Construction Technique
„ The graphical procedure involves the
location of a point P which lies on the curve
determined by the tangents and points
described.
General Graphical Construction Technique
The graphical method is as follows:
1. Draw the line BE and OF through D.
2. Draw any radial line AG through A.
3. AG intersects OF at J and BE at H.
4. Draw the line OK through H and the line BL through J.
5. The lines OK and BL intersect at P.
6. Then P is the required point which lies on the specified
curve.
Analytic Approach
Analytic Approach
Analytic Approach:
1. Obvious Approach
2. Control Conditions
Analytic Approach (Obvious Approach )
„
A more obvious approach was to make the
conic Ax2+Bxy+Cy2+Dx+Ey+F=0 pass
through five given points (no three collinear)
by substituting in the equation the
coordinates of the five points in turn.
„
This give five simultaneous equations to
solves for the five essential ration.
Analytic Approach (Control Conditions)
„
Conic control conditions most
commonly used in designing an aircraft
streamline shape.
„
Two point-slopes (for four conditions)
and a control point (fifth condition).
Analytic Approach (Control Conditions)
„
All the conics of the family have the
same tangents t1 and t2 at O and B
respectively.
Analytic Approach (Control Conditions)
„
Every conic which is tangent to t1 at O
and t2 at B is uniquely determined by a
third point D and therefore belongs to
the family.
Analytic Approach (Control Conditions)
„
There are two distinct degenerate
conics in the family; the tangents t1 and
t2 form the one, and the chord of
contact (t3) (taken twice) constitutes the
other.
Analytic Approach (Control Conditions)
„
„
If : t1=0, t2=0 and t3=0
Then the equation of the family of conics
becomes:
t t + kt = 0
2
1 2
3
tt
k =−
t
1 2
2
3
Analytic Approach (Control Conditions)
„
To develop the equation of any particular
conic of the family, it is merely necessary
to evaluate k for the coordinates of a
point D (control point).
Analytic Approach (Control Conditions)
O: (0,0), A: (a,0), B: (b,c), D: (d,e)
Tangent OA: y=0
Tangent BA: (a-b)y+c(x-a)=0
Chord OB: cx-by=0
Analytic Approach (Control Conditions)
Tangent OA: y=0
Tangent BA: (a-b)y+c(x-a)=0
Chord OB: cx-by=0
„
tt
k =−
t
1 2
2
3
Through substitution of tangents and chord equations in k
equation:
y[(a − b) y + c( x − a )] + k (cx − by ) = 0
2
[
(c − e)a − (cd − be)]e
k=
(cd − be)
2
Analytic Approach (Control Conditions)
„
By a simple reduction, expressed as y=f(x):
y = Px + Q ± Rx + Sx + T
2
0.25 − ak
m
bk − 0.5
P=
m
⎛a⎞
S = P⎜ ⎟
⎝m⎠
⎛a⎞
Q = 0.5⎜ ⎟
⎝m⎠
T =Q
R=
2
2
(a − b + kb )
m=
c
2
Analytic Approach (Control Conditions)
„
Slope equation:
dy
2 Rx + S
= P−
dx
2 Rx + Sx + T
2
Analytic Approach (Control Conditions)
„
The invert form of equation: the form x=f(y):
x = Py + Ry + Sy
2
[
(c − e)a − (cd − be)]e
k=
(cd − be)
2
2bk − 1
P=
2n
1 − 4ak
R=
n = ck
( 2n)
a
S=
n
2
Analytic Approach (Control Conditions)
„
Example: O: (0,0), A: (5,0), B: (6,3), D: (4,1)
1
11
1
5
5
25
k = , m = 1, , R = − , P = , S = , Q = , T =
9
36
6
6
2
4
1
11
y = f ( x) = x + 5 2 − − x + 5 6 x + 25 4
6
36
2
1
1
11
1
k = , n = , P = , R = − , S = 15
4
9
3
2
1
11
x = f ( y ) = y + − y + 15 y
2
4
2
Application of Large Scale
Digital Computer Technique
Two Tests
„
Two special tests are significant enough to have
warranted programming development.
1. For a parabolic section, awareness of which “flag”
the computing process thru a greatly simplified, time
saving subroutine
2. For a hyperbolic section with vertical asymptote; loss
of significance is avoided by diverting the algebraic
through a special set of formulas.
Application of Large Scale
Digital Computer Technique
„
„
„
We note five conditions defining the conic: two point
slopes (for 4 conditions) and a control point (fifth
condition)
To minimize algebraic complexities: the equation origin
at one of the end points.
Assuming that neither tangent is parallel (or
perpendicular ) to either coordinate axis.
Application of Large Scale
Digital Computer Technique
y = Px + Q ± Rx + Sx + T
2
„
Let:
C = ad − bc
E = af − be
J = cf − de
„
Then: k = (C − E + J ) E
J
2
Application of Large Scale
Digital Computer Technique
(C − E + J ) E
k =
J
2
0.25 − k
R=
N
0 .5 a
Q=
N
T =Q
ck − 0.5a
S=
N
cdk + a (b − d ) + 0.5C
P=
NC
2
2
2
c k + a (a − c)
N=
C
2
„
Application of Large Scale
Digital Computer Technique
If the program tests and finds the conic to be a
parabola:
R=0
2Ca
Q=
(c − 2 a )
T =Q
4C
S=
(c − 2 a )
d − 2b
P=
c − 2a
2
2
2
3
y = Px + Q ± Sx + T
Application of Large Scale
Digital Computer Technique
„
If c=2a :
ad − bc
R=
ac
b
S=
a
2
y = Rx + Sx
2
„
Application of Large Scale
Digital Computer Technique
If N=0, ( a ≠ 0) , hyperbola section:
bP
R = 1+
a
2
ac
Q=
Rx
Sx
+
C (c − a )
y=
Px
+
Q
bQ
S=
a
ac(c − 2a )
P=
C (c − a )
2
2
Application to a
specific Fuselage
Application to a specific Fuselage
„
Application to a specific Fuselage
The lower right hand quadrant of the fuselage
cross section: O: (0,0), A: (a,0), B: (a,c), D: (d,e)
Tangent OA: y=0
Tangent AB: c(x-a)=0
Chord OB: cx-ay=0
Application to a specific Fuselage
ec(a − d )
2
k=
y = Px + Q − Rx + Sx + T
(cd − ae)
2
2ack − c
P=
2a k
ac
Q=
2a k
c − 4ac k
R=
4( a k )
S = 2 PQ
2
2
2
2
2
T =Q
2
2
dy
Rx + S 2
= P−
Rx + Sx + T
dx
2
Application to a specific Fuselage
„
Actual Numeric:
O: (0,0), A: (31,0), B: (31,36), D: (15,2.34662)
ec(a − d )
k=
(cd − ae)
2
2ack − c
2a k
ac
Q=
2a k
c − 4ac k
R=
4( a k )
S = 2 PQ
T =Q
P=
2
2
2
2
2
2
2
k = 0.0061909552
P = −1.86416997
Q = 93.789269
R = 2.126534465
S = −349.6782776
T = 8796.426980
y = Px + Q − Rx + Sx + T
2
y = −1.86417x + 93.7893 − 2.1265345x − 349.678278x + 8796.42698
2