DIFFERENTIAL GEOMETRY AND RELATIVITY
DX ? ? X=F(A) = DF DX X=A OR F ?1?(F(A)) = DS = ???(S?1(S)) S?(S?1(S)) = ???(T) S?(T) = ???(T) ????(T)? NOTICE D?? DS = ???(S) IS A UNIT VECTOR IN THE DIRECTION OF THE VELOCITY VECTOR OF ??(T) DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER ALSO, ?T(S) = ???(S) = , B NOTICE THAT ?T = ???(T) ????(T)? = ???(S?1(S)) ????(S?1(S))? DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER IF A > 0 DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER NOTICE THAT IN TERMS OF T, ?N(T) = ?(COS T, SIN T, 0) THAT IS, ?N(T) IS A VECTOR THAT POINTS FROM THE PARTICLE AT ??(T) = (A COS T, A SIN T, BT) BACK TO THE Z?AXIS (THAT IS, ?N(T) IS A UNIT VECTOR FROM ??(T) TO (0, 0, BT)) NOTE IF WE TAKE B = 0 IN EXAMPLE 3, WE JUST GET ??(T) TO TRACE OUT A CIRCLE OF RADIUS A IN THE XY?PLANE THE CURVATURE OF THIS CIRCLE IS K(S) = A A2 + B2 = A THEREFORE, CIRCLES OF “SMALL” RADIUS HAVE “LARGE” CURVATURE AND CIRCLES OF “LARGE” RADIUS HAVE “SMALL” CURVATURE (AND THE CURVATURE OF A STRAIGHT LINE IS 0) SEE FIGURE I-4 DE?NITION FOR A GIVEN VALUE OF S, THE CIRCLE OF RADIUS K(S) WHICH IS TANGENT TO ?? AND WHICH LIES IN THE PLANE OF ?T(S) AND ?N(S) IS THE OSCULATING CIRCLE OF ?? AT POINT ??(S) THE CENTER OF THE OSCULATING CIRCLE IS THE CENTER OF CURVATURE OF ?? AT POINT ??(S), DENOTED ?C(S) THE PLANE CONTAINING THE OSCULATING CIRCLE IS THE OSCULATING PLANE SEE FIGURE I-6 NOTE ?C(S) IS CALCULATED BY GOING FROM POINT ??(S) A DISTANCE K(S) IN THE DIRECTION ?N(S) THAT IS, ?C(S) = ??(S) + K(S) ?N(S) EXAMPLE (P 13, # 12) CONSIDER THE HELIX ABOVE PARAMETERIZED IN DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER FIND ?C(S) AND SHOW THAT IT IS ALSO A HELIX SOLUTION THE CENTER OF CURVATURE IS ?C(S) = ??(S) + A AND T = S ? A2 + B2, THEN ?C(T) = (A COS T, A SIN T, BT) , WHICH IS A CIRCULAR HELIX NOTE THE CURVATURE K(S) OF A CURVE ??(S) GIVES AN IDEA OF HOW A CURVE “TWISTS” BUT DOES NOT PROVIDE A COMPLETE DESCRIPTION OF THE CURVES “GYRATIONS” (AS THE TEXT PUTS IT - SEE PAGE 8) THERE IS INFORMATION IN HOW THE OSCULATING PLANE TILTS AS S VARIES DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER RECALL A PLANE IN E3 IS DETERMINED BY A POINT (X0, Y0, Z0) AND A NOR- MAL VECTOR ?N = (A, B, C) (WE WILL NOT NOTATIONALLY DISTINGUISH BETWEEN POINTS AND VECTORS) IF (X, Y, Z) IS A POINT IN THE PLANE, THEN A VECTOR FROM (X0, Y0, Z0) TO (X, Y, Z) IS PERPENDICULAR TO ?N AND SO ?N (X ?X0, Y ?Y0, Z ?Z0) = (A, B, C) (X ?X0, Y ?Y0, Z ?Z0) = A(X ?X0) + B(Y ?Y0) + C(Z ?Z0) = 0 THIS CAN BE REARRANGED AS AX + BY + CZ = D FOR SOME CONSTANT D NOTICE THAT “TWISTINGS” OF THE PLANE WOULD BE RE?ECTED IN CHANGES IN THE DIRECTION OF THE NORMAL VECTOR EXAMPLE FIND THE EQUATION OF THE PLANE THROUGH THE POINTS (1, 2, 3), (?2, 3, 3) AND (1, 2, 4) SOLUTION THE VECTORS ?A = (1 ?(?2), 2 ?3, 3 ?3) = (3, 1, 0) AND ?B = (1 ?1, 2 ?2, 4 ?3) = (0, 0, 1) BOTH LIE IN THE DESIRED PLANE RECALL THAT IN E3, ?A AND ?B ARE BOTH ORTHOGONAL TO ?A ×?B (PROVIDED ?A IS NOT A SCALAR MULTIPLE OF ?B - SEE APPENDIX A FOR MORE DETAILS) SO WE CAN TAKE ?N = ?A ×?B AS A NORMAL VECTOR FOR THE DESIRED PLANE ?N = ?A ×?B = ?I ?J ?K =?I ?3?J + 0?K = (1, ?3, 0) SO THE DESIRED PLANE SATIS?ES 1(X ?1) ?3(Y ?2) + 0(Z ?3) = 0 OR X ?3Y = ?5 EXPRESSED PARAMETRICALLY, X = ?5 + 3T DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER Y = T Z = Z (A “FREE VARIABLE”) (AGAIN, WE MAKE NO NOTATIONAL DISTINCTION BETWEEN A VECTOR AND A POINT THERE IS A DI?ERENCE, THOUGH: POINTS HAVE LOCATIONS, BUT VECTORS DON’T [NONZERO VECTORS HAVE A LENGTH AND A DIRECTION, BUT NO POSITION] ) DE?NITION THE BINORMAL VECTOR IS ?B = ?T × ?N NOTE ?B IS ORTHOGONAL TO BOTH ?T AND ?N (AND THEREFORE TO THE OSCULATING PLANE) THE DERIVATIVE OF ?B IS ?B? = (?T × ?N)? = ?T ? × ?N + ?T × ?N? (WHERE ? REPRESENTS DERIVATIVE WITH RESPECT TO WHATEVER THE VARIABLE OF PARAMETERIZATION IS) SINCE ?T ? = K ?N, ?T ? × ?N = ?0 AND SO ?B? = ?T × ?N? SINCE ?N IS A UNIT VECTOR, ?N? IS PERPENDICULAR TO ?N (AS ARGUED ABOVE FOR ?T) ALSO, ?T IS PERPENDICULAR TO ?N SINCE ?N = (1/K)?T ? SO BOTH ?T AND ?N? ARE PERPENDICULAR TO ?N AND SO ?B? = ?T × ?N? IS A MULTIPLE OF ?N (SINCE WE ARE IN 3-DIMENSIONS), SAY ?B? = ?? ?N (NOTICE THAT IF ?T AND ?N? ARE MULTIPLES OF EACH OTHER, THEN ? = 0 ) DE?NITION THE TORSION OF ?? AT ??(S) IS THE FUNCTION ?(S) WHERE ?B?(S) = ??(S) ?N NOTE THE TORSION MEASURES THE TWISTING (OR TURNING) OF THE OSCULATING PLANE AND THEREFORE DESCRIBES HOW MUCH ?? “DEPARTS FROM BEING A PLANE CURVE” (AS THE TEXT SAYS - SEE PAGE 9) DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER EXAMPLE CALCULATE ?B, ?B? AND ?(S) FOR THE HELIX ABOVE SOLUTION WE HAVE ?T(S) = , SO ?B = ?T × ?N = THEREFORE, SINCE ?B? = ?? ?N, WE HAVE ?(S) = B A2 + B2 NOTE NOTICE THAT THE TORSION IS A CONSTANT IN THE PREVIOUS EXAMPLE THIS MAKES SENSE SINCE THE OSCULATING PLANE TILTS AT A CONSTANT RATE AS A PARTICLE TRAVELS (UNIFORMLY) UP THE HELIX (OR “SPRING”) WITH B = 0, THE HELIX IS, IN FACT, A CIRCLE IN THE XY?PLANE AND SO THE OSCULATING PLANE DOES NOT CHANGE AND ? = 0 (IF ?(S) = 0 FOR ALL S, THEN ??(S) IS PLANAR - SEE PAGE 13 #13 ) NOTE WE WILL SEE THAT THE SHAPE OF A CURVE IS COMPLETELY DETERMINED BY THE CURVATURE K(S) AND TORSION ?(S) DIFFERENTIAL GEOMETRY (AND RELATIVITY) - SUMMER