Applied Mathematics
Vol.08 No.09(2017), Article ID:79281,16 pages
10.4236/am.2017.89097
Projection of the Semi-Axes of the Ellipse of Intersection
P. P. Klein
Computing Center, University of Technology Clausthal, Clausthal-Zellerfeld, Germany
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 22, 2017; Accepted: September 22, 2017; Published: September 25, 2017
ABSTRACT
It is well known that the line of intersection of an ellipsoid and a plane is an ellipse (see for instance [1] ). In this note the semi-axes of the ellipse of intersection will be projected from 3d space onto a 2d plane. It is shown that the projected semi-axes agree with results of a method used by Bektas [2] and also with results obtained by Schrantz [3] .
Keywords:
Ellipsoid and Plane Intersection, Projection of the Semi-Axes of the Ellipse of Intersection
1. Introduction
Let an ellipsoid be given with the three positive semi-axes , ,
(1)
and a plane with the unit normal vector
which contains an interior point of the ellipsoid. A plane spanned by vectors , and containing the point is described in parametric form by
(2)
Inserting the components of into the equation of the ellipsoid (1) leads to the line of intersection as a quadratic form in the variables t and u. Let the scalar product in for two vectors and be denoted by
and the norm of vector by
With the diagonal matrix
the line of intersection has the form:
(3)
As is an interior point of the ellipsoid the right-hand side of Equation (3) is positive.
Let and be unit vectors orthogonal to the unit normal vector of the plane
(4)
(5)
and orthogonal to eachother
(6)
If vectors and have the additional property
(7)
the matrix in (3) has diagonal form. If condition (7) does not hold for vectors and , it can be fulfilled, as shown in [1] , with vectors and obtained by a transformation of the form
(8)
with an angle according to
(9)
Relations (4), (5) and (6) hold for the transformed vectors and instead of and . If plane (2) is written instead of vectors and with the transformed vectors and the matrix in (3) has diagonal form because of condition (7):
Then the line of intersection reduces to an ellipse in translational form
(10)
with the center
(11)
and the semi-axes
(12)
where
(13)
Because of the numerator in (12) is positive.
Putting
(14)
the semi-axes A, B given in (12) can be rewritten as
(15)
In [1] it is shown that and according to (14) are solutions of the following quadratic equation
(16)
Furthermore it is proven in [1] that d according to (13) satisfies
(17)
2. Projection of the Ellipse of Intersection onto a 2-d Plane
The curve of intersection in 3d space can be described by
(18)
with center , where and are from (11), semi-axes A and B from (12), and vectors and obtained after a suitable rotation (8) starting from initial vectors and (see for instance [1] ).
Without loss of generality the plane of projection of the ellipse (18) shall be the plane. The angle between the plane of intersection (2) containing the ellipse (18) and the plane of projection is denoted by . The same angle is to be found between the unit normal of the plane of intersection (2) and the -direction, normal to the plane of projection. Denoting the unit vector in -direction by the definition of the scalar product (see for instance [4] ) yields
(19)
where holds for .
Let us assume that the plane of intersection (2) is not perpendicular to the
plane of projection, the plane. This means that is valid and
according to (19) holds.
The ellipse of intersection (18) projected from 3d space onto the plane has the following form:
(20)
In general the two dimensional vectors and are not orthogonal because their orthogonality in 3d space implies
which need not be zero. In order to calculate the lenghts of the semi-axes A and B projected from 3d space onto the plane the following linear system deduced from (20) with the abbreviations and is treated:
(21)
The determinant of the linear system (21), , is different from zero. This can be shown by noting that is the third component of the vector . At first this vector is not affected by rotation (8):
This result was obtained by applying the rules for the cross product in . Furthermore one obtains employing the Grassman expansion theorem (see for instance [4] ):
because of and . Thus one ends up with
(22)
which is positive because of (19) for angles with .
Solving the linear system (21) leads to
Since together with (22) the following quadratic equation in and is obtained:
Expanding the squares on the left side and using the denotations
(23)
arranged as a matrix
(24)
leads to
(25)
as a real symmetric matrix can be diagonalized and thus is similar to the diagonal matrix of its eigenvalues , :
with a nonsingular transformation matrix , being orthogonal, i.e. , the inverse of is equal to the transpose of . Putting
the quadratic equation (25) in reduces to
(26)
The eigenvalues , are positive because is positive definite; this is true since the terms and are positive. For this is clear; for the second term, the determinant of , holds because of (22):
(27)
Dividing (26) by yields
This is an ellipse projected from 3d space (18) onto the plane with the semi-axes
(28)
With (19) one obtains from (28)
(29)
3. Calculation of Semi-Axes According to a Method Used by Bektas
Let the ellipsoid (1) be given and a plane in the form
(30)
The unit normal vector of the plane is:
(31)
The distance between the plane and the origin is given by
(32)
The plane written in Hessian normal form then reads:
Without loss of generality shall be assumed. Then holds:
Forming and substituting into equation (1) gives:
(33)
with
(34)
In the sequel the determinant of the following matrix will be needed:
(35)
In order to get rid of the linear terms and in (33) the following translation can be performed: , with parameters h and k to be determined later. After substitution into (33) one obtains:
(36)
The terms and in (36) vanish if h and k are determined by the linear system:
(37)
The linear system (37) has as matrix of coefficients, the determinant of which is given in (35). It is nonzero because of the assumption . Solving the linear system (37) yields:
(38)
Substituting the terms (34) into (38) gives the result:
(39)
With the terms h and k from (39) the constant term in (36) turns out to be, together with (17):
Thus the quadratic equation (36) reduces to:
(40)
as a real symmetric matrix can be diagonalized and thus is similar to the diagonal matrix of its eigenvalues , :
with a nonsingular transformation matrix , being orthogonal, i.e. , the inverse of is equal to the transpose of . Putting
the quadratic equation (40) in reduces to
(41)
The eigenvalues , are positive because is positive definite; this is true since the terms and are positive. For this is clear; the second term, the determinant of , is given in (35). If a point of the plane (30) exists which is an interior point of the ellipsoid (1), then is positive (see Section 1). Dividing (41) by yields
This is an ellipse in the plane with the semi-axes
(42)
4. Calculation of Projected Semi-Axes According to Schrantz
In [3] the ellipse
(43)
with the semi-axes A and B is projected from plane E onto plane . As in
Section 2 the angle between the two planes is denoted by , with . Let , with , be the angle between the major axis of the original
ellipse (43) and the straight line of intersection of the two planes E and
and let be a phase-shift with and where
the angles and are determined by
(44)
The projected ellipse in the plane is given by
(45)
with
(46)
Eliminating parameter from (45) yields a quadratic equation in and
or written with the elements
(47)
forming matrix
one obtains
(48)
as a real symmetric matrix can be diagonalized and thus is similar to the diagonal matrix of its eigenvalues , :
with a nonsingular transformation matrix , being orthogonal, i.e. , the inverse of is equal to the transpose of . Putting
the quadratic equation (48) in reduces to
(49)
The eigenvalues , are positive, if G is positive definite; this is the case if the terms and are positive. For this is true; the second term, the determinant of G, given by
(50)
is positive for . Dividing (49) by for yields
This is an ellipse in the plane with the semi-axes
(51)
5. Some Auxiliary Means
Let stand for the following matrix:
(52)
and be a place holder for the matrices and used above. The semi-axes , projected onto the plane, given in (28), are compared with the semi-axes , . It will be shown that the two polynomials
(53)
have the same coefficients and thus have the same zeros:
(54)
In the first step will be proven. In the second step
(55)
will be shown. This is sufficient, since by adding to both sides of (55) one obtains:
which yields since the semi-axes are positive.
, are the zeros of the characteristic polynomial of . This can be expressed in two ways:
Comparing the coefficients one obtains
(56)
Similarly the results for matrix instead of are
(57)
6. Comparison of the Semi-Axes AL, BL with AM, BM
In the first step will be proven. According to (28) and (42) holds:
(58)
(59)
In the case of matrix combining (56) and (27) yields:
(60)
In the case of matrix combining (57), where is substituted for ,and (35) leads to:
(61)
Because and are solutions of (16)
(62)
holds and because of (60), (15), (62) and (61)
(63)
Thus with (58), (60), (63) and (59) one concludes
In the second step because of (28) and (60) holds
(64)
Because of (42), (61) and (62) holds
(65)
Together with
(66)
(65) yields
(67)
In continuation of (64), because and are fulfilling (4) and (5), the following relations hold:
(68)
with
(69)
because and are solutions of (16). Combining (64), (68), (69) and (67) one obtains:
(70)
To simplify the term in round brackets of (70) the following relations are used:
because of (see Section 2), and
according to (14). The term in round brackets of (70) thus becomes:
because and have been chosen in such a way that condition (7) is fulfilled.
7. Comparison of the Semi-Axes AL, BL with AG, BG
In the first step will be proven. According to (29) and (51) holds:
(71)
(72)
In the case of matrix combining (56), (27) and (19) yields:
(73)
In the case of matrix combining (57), where is substituted for ,and (50) leads to:
(74)
Substitution of (73) into (71) and (74) into (72) yield
(75)
According to the definition of given in the beginning of Section 4 together with (44) and (46) one obtains:
Substituting this into (75) one ends up with .
In the second step because of (64), (56) and (23) holds
(76)
Because of (51), (74), (57), where matrix is substituted for matrix ,and (47) holds
(77)
(77) is continued by substituting and from (46)
(78)
Comparing (76) and (78), in order to show equality , it has to be proven:
(79)
As already described in the beginning of Section 4 the ellipse (43) is projected from the original plane onto the plane . Both planes are forming an
angle with . Without loss of generality the intersection of
and , , shall be the -axis of the coordinate system in plane . The original plane thus contains the following three points: , , and can therefore be described by the following equation:
(80)
The unit normal vector of plane (80) given by (31) is
(81)
In order to describe a unit vector in the plane the equations (4) must hold:
(82)
The second equation of (82) yields . Substituting this into the first equation of (82) results in:
or
(83)
If the unit vector is forming the angle with the -axis and is designating a unit vector in -direction according to the definition of the scalar product (see for instance [4] ) holds
From (83) one obtains
yielding and furthermore with the first equation of (82) . From
and one obtains
By transformation (8) one obtains
Thus equation (79) turns into
(84)
Equation (84) is fulfilled if holds. The -case leads to , which means that (84) is fulfilled if transformation (8) is the identity, i.e. , ; the -case leads to , meaning that if , the angle between the
major axis of the ellipse (43) and the -axis, is chosen to be then (84) is true.
8. Numerical Example
The following numerical example is taken from [2] . Let the semi-axes of the ellipsoid (1) be
and let the plane be given by
The following calculations have been performed with Mathematica. According to (31) the unit normal vector of the plane is
Furthermore in (32) the distance of the plane to the origin is given
According to (17) d can be calculated.
Starting with an arbitrary unit vector orthogonal to the unit normal vector , for instance
calculating to be orthogonal to both according to and, as , perform a rotation with angle given in (9), yielding new vectors and according to (8), which are plugged into and .
The semi-axes A and B in 3d space according to (12) can be calculated to be
Furthermore having calculated the eigenvalues and the semi-axes and projected onto the plane according to (28) are
The same results are obtained calculating and according to (42) by the method used by Bektas.
9. Conclusion
The intention of this paper was, to show that the semi-axes of the ellipse of intersection projected from 3d space onto a 2d plane are the same as those calculated by a method used by Bektas. Furthermore they are also equal to the semi-axes of the projected ellipse obtained by Schrantz.
Cite this paper
Klein, P.P. (2017) Projection of the Semi-Axes of the Ellipse of Intersection. Applied Mathematics, 8, 1320-1335. https://doi.org/10.4236/am.2017.89097
References
- 1. Klein, P.P. (2012) On the Ellipsoid and Plane Intersection Equation. Applied Mathematics, 3, 1634-1640. https://doi.org/10.4236/am.2012.311226
- 2. Bektas, S. (2016) On the Intersection of an Ellipsoid and a Plane. International Journal of Research in Engineering and Applied Sciences, 6, 273-283.
- 3. Schrantz, G.R. (2004) Projections of Ellipses and Circles. Hamline University. (To be found in the internet)
- 4. Bronshtein, I.N., Semendyayev, K.A., et al. (2007) Handbook of Mathematics. 5th Edition, Springer-Verlag Berlin Heidelberg.