MECH 576 Geometry in Mechanics September 16, 2009 Using Line Geometry 1 Deriving Equations in Line Coordinates Four exercises in deriving fundamental geometric equations with line coordinates will be conducted. These results may be widely used to formulate constraint equations to solve many useful geometric problems. • First the derivation of the summations, introduced earlier, 3 3 X X p = Pa ∩ P : Pi = Pijpj,P = Pr ∩ p : pi = pijPj (1) j=0 j=0 that define the plane p on a given axial line Pa and point P are derived using the plane equation Grassmannian. This is followed by inductive reasoning to express the dual relation, i.e., the point P on given radial line Pr and plane p. • Next the set of all lines tangent to a given sphere is derived. This is a simple second order complex; a three parameter set of lines. • Finally the cylinder and cone of revolution, using a given tangent cylinder and an absolute point on the axis and a Euclidean point on the apex, respectively, are expressed as a one parameter set of ruling line coordinates. These are converted to point form with any two constraint equations, selected from the first set of four expressed by Eq. 1, and setting all Pi = 0. 2 Spanning Plane and Piercing Point Consider a plane, given by its four homogeneous coordinates, p{P0 : P1 : P2 : P3} and a line, given by its six homogeneous radial Pl¨ucker coordinates, Pr{p01 : p02 : p03 : p23 : p31 : p12} Recall the identical axial line is related as Pa{P01 : P02 : P03 : P23 : P31 : P12} = λ{p23 : p31 : p12 : p01 : p02 : p03}, λ 6= 0 1 The given plane and line, if linearly independent, intersect on a point. The purpose here is to illustrate how to structure the 4×4 Grassmannian determinant form of the standard point equation so as to produce the well known sum of products piercing point equation 3 X pi = pijPj j=0 to yield the four homogeneous coordinates of the point P {pi}, i = 0,..., 3,P = Pr ∩ p Recall that the conventions, pij = 0 if i = j and pij = −pji if i 6= j, apply in this summation. Imagine that Pr is expressed by some plane pair q{Q0 : Q1 : Q2 : Q3}, r{R0 : R1 : R2 : R3} whose coordinates may be expanded to axial Pl¨ucker coordinates. Pr ≡ Pa{Q0R1 − R0Q1 : Q0R2 − R0Q2 : Q0R3 − R0Q3 : Q2R3 − R2Q3 : Q3R1 − R3Q1 : Q1R2 − R1Q2} So the point equation determinant is written X0 X1 X2 X3 P P P P 0 1 2 3 = 0 Q0 Q1 Q2 Q3 R0 R1 R2 R3 Expanding on top row minors computes the piercing point homogeneous coordinates as coeffi- cients. But in this case each coefficient is computed not by directly evaluating the 3 × 3 cofactor determinant but by expanding on its top row minors. These are the remaining second row entries of the original 4 × 4 after crossing out the top row minor’s row and column. The first top row minor and cofactor are P P P 1 2 3 + Q1 Q2 Q3 X0 ⇒ R1 R2 R3 which expands as [(Q2R3 − R2Q3)P1 − (Q1R3 − R1Q3)P2 + (Q1R2 − R1Q2)P3]X0 ⇒ which is the same as [(Q2R3 − R2Q3)P1 + (Q3R1 − R3Q1)P2 + (Q1R2 − R1Q2)P3]X0 ⇒ and this becomes, via the substitution of the appropriate axial Pl¨ucker coordinates [P23P1 + P31P2 + P12P3]X0 ⇒ 2 or via the substitution of the appropriate radial Pl¨ucker coordinates [p01P1 + p02P2 + p03P3]X0 ⇒ Finally, this compares identically to the first row of the summation where i = 0 and p00 = 0 and p0, the first coefficient of the point equation, is obtained. p0 = p00P0 + p01P1 + p02P2 + p03P3 The line-point-plane relation is clearly understood by seeing through to the end the computation of the next three point equation coefficients. The second top row minor and cofactor are P P P 0 2 3 − Q0 Q2 Q3 X1 ⇒ R0 R2 R3 which expands as −[(Q2R3 − R2Q3)P0 − (Q0R3 − R0Q3)P2 + (Q0R2 − R0Q2)P3]X1 ⇒ and this becomes, via the substitution of the appropriate axial Pl¨ucker coordinates −[P23P0 − P03P2 + P02P3]X1 ⇒ or via the substitution of the appropriate radial Pl¨ucker coordinates −[p01P0 − p12P2 + p31P3]X1 ⇒ Finally, this compares identically to the first row of the summation where i = 1, pij = −pji and p11 = 0 and p1, the second coefficient of the point equation, is obtained. p1 = p10P0 + p11P1 + p12P2 + p13P3 The third top row minor and cofactor are P P P 0 1 3 + Q0 Q1 Q3 X2 ⇒ R0 R1 R3 which expands as [(Q1R3 − R1Q3)P0 − (Q0R3 − R0Q3)P1 + (Q0R1 − R0Q1)P3]X2 ⇒ and this becomes, via the substitution of the appropriate axial Pl¨ucker coordinates [−P31P0 − P03P1 + P01P3]X2 ⇒ or via the substitution of the appropriate radial Pl¨ucker coordinates [−p02P0 − p12P1 + p23P3]X2 ⇒ 3 Finally, this compares identically to the first row of the summation where i = 2, pij = −pji and p22 = 0 and p2, the third coefficient of the point equation, is obtained. p2 = p20P0 − p21P1 + p22P2 + p23P3 The fourth top row minor and cofactor are P P P 0 1 2 − Q0 Q1 Q2 X3 ⇒ R0 R1 R2 which expands as −[(Q1R2 − R1Q2)P0 − (Q0R2 − R0Q2)P1 + (Q0R1 − R0Q1)P1]X3 ⇒ and this becomes, via the substitution of the appropriate axial Pl¨ucker coordinates −[P12P0 − P02P1 + P01P2]X3 ⇒ or via the substitution of the appropriate radial Pl¨ucker coordinates −[p03P0 − p31P1 + p23P2]X3 ⇒ Finally, this compares identically to the first row of the summation where i = 3, pij = −pji and p33 = 0 and p3, the fourth coefficient of the point equation, is obtained. p3 = p30P0 + p31P1 + p32P2 + p33P3 This completes the elaboration of the intersection of a projective line and plane. It is not necessary to do the same for the intersection of a projective line and point to get the projective plane so defined. This is because one may invoke the principle of duality and exchange point with plane coordinates and radial line coordinates with axial ones. Consider now the degenerate cases, i.e., if the given elements are linearly dependent. This means that the piercing point does not exist because the line is on the plane or the spanning plane does not exist because the point is on the line. All four coordinates vanish, pi = 0 or Pi = 0. This is a very useful way to express that P ∈ P or p ∈ P and will be used to derive implicit equations of ruled surfaces which are easily formulated using line geometric concepts. 3 The Spherical Tangent Line Complex The set of all lines T{t01 : t02 : t03 : t23 : t31 : t12} tangent to a given sphere, radius R, centred on point M{m0 : m1 : m2 : m3} can be deduced by equating the square of the moment magnitude of a typical tangent line’s direction numbers, about the point M, obtained in two different ways. First, the square of the difference between the typical tangent line’s moment t23 : t31 : t12 and the moment of a line on M which has the direction numbers t01 : t02 : t03 of the typical tangent line. This can be expressed as 2 1 2 t23 − (m2t03 − m3t02) t23 m1 t01 m0 1 1 t − m × t = t31 − (m3t01 − m1t03) 31 2 02 m0 m0 1 t12 m3 t03 t12 − (m1t02 − m2t01) m0 4 Then the square of the moment of the typical line’s direction numbers about M is formed directly as 2 t01 R t02 t03 2 Therefore the desired equation of the line complex is the difference multiplied by m0. 2 2 [m0t23 − (m2t03 − m3t02)] + [m0t31 − (m3t01 − m1t03)] 2 2 2 2 2 +[m0t12 − (m1t02 − m2t01)] − (m0R) (t01 + t02 + t03) = 0 The Spherical Tangent Line Complex (a 3-parameter set of lines described by a 2nd order equation in Pleucker coordinates) z (MECH576)STLC y o O x Figure 1: Complex Geometry 4 Cylinder of Revolution Beginning with the equation of the tangent line complex on the sphere 2 2 [m0t23 − (m2t03 − m3t02)] + [m0t31 − (m3t01 − m1t03)] 2 2 2 2 2 +[m0t12 − (m1t02 − m2t01)] − (m0R) (t01 + t02 + t03) = 0 5 consider a cylinder of revolution of radius R and whose axis is line A{a01 : a02 : a03 : a23 : a31 : a12} Note that if the axial direction is given as on M, the sphere centre, and A{0 : a1 : a2 : a3}, an absolute point then A is immediately available as A = A ∩ M. If your sensibilities are offended by the use of ∩, intersection rather than union, consider that in the space of dual, plane coordinates the these two points assume plane-like form. First the line complex is converted to the surface equation in terms of typical ruling lines P{p01 : p02 : p03 : p23 : p31 : p12} by noting that t0j = a0j and moments mit0j − mjt0i = aij and making these substitutions to yield 2 2 2 2 2 2 2 (p23 − a23) + (p31 − a31) + (p12 − a12) − R (a01 + a02 + a03) = 0 This becomes a one parameter set of lines when combined with the Pl¨ucker condition, in this case a linear equation. a01p23 + a02p31 + a03p12 = 0 The point equation is produced by using two elements of the expansion P ∈ P which gives the degenerate plane defined by a line and a point on it. 3 X Pi = Pijpj = 0 j=0 Substituting radial for axial line coordinates, the cylinder is given by 2 2 (a23p0 + a02p3 − a03p2) + (a31p0 + a03p1 − a01p3) 2 2 2 2 2 +(a12p0 + a01p2 − a02p1) − R (a01 + a02 + a03) = 0 The obvious advantage of this formulation is that it gives one the option to write the implicit form of this quadric surface directly from its geometric definition, without resort to coordinate transformation or even trigonometric functions.