222 - Topic 01 - Postion, Velocity, Acceleration

# 3-D motion and vector calculus # Copyright 2009, Robert A. Beezer # Creative Commons BY-SA 3.0 US # # # 2009/02/15 Built on Sage 3.3.rc0 # 2009/02/17 Improvements from Jason Grout # # variable parameter is t # later at a particular value named t0 # # un-comment double hash (##) to get # time-consuming torsion computation # var('t') assume(t, 'real') # # parameter range # start=-4*pi stop=8*pi # # position vector definition # edit here for new example # example is wide ellipse # adjust figsize in final show() to get accurate aspect ratio # a=1/(8*pi) c=(3/2)*a position=vector( (exp(a*t)*cos(t), exp(a*t)*sin(t), exp(c*t)) ) # # graphic of the motion itself # path = parametric_plot3d( position.list(), (t, start, stop), color = "black" ) # # derivatives of motion, lengths, unit vectors, etc # velocity = derivative( position, t) acceleration = derivative(velocity, t) speed = velocity.norm() speed_deriv = derivative(speed, t) tangent = (1/speed)*velocity dT = derivative(tangent, t) normal = (1/dT.norm())*dT binormal = tangent.cross_product(normal) ## dB = derivative(binormal, t) # # interact section # slider for parameter, 24 settings # checkboxes for various vector displays # computations at one value of parameter, t0 # @interact def _(t0 = slider(float(start), float(stop), float((stop-start)/24), float(start) , label = "Parameter"), pos_check = ("Position", True), vel_check = ("Velocity", False), tan_check = ("Unit Tangent", False), nor_check = ("Unit Normal", False), bin_check = ("Unit Binormal", False), acc_check = ("Acceleration", False), tancomp_check = ("Tangential Component", False), norcomp_check = ("Normal Component", False) ): # # location of interest # pos_tzero = position(t=t0) # # various scalar quantities at point # speed_component = speed(t=t0) tangent_component = speed_deriv(t=t0) normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 ) curvature = normal_component/speed_component^2 ## torsion = (1/speed_component)*(dB(t0)).dot_product(normal(t0)) # # various vectors, mostly as arrows from the point # pos = arrow3d((0,0,0), pos_tzero, rgbcolor=(0,0,0)) tan = arrow3d(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) ) vel = arrow3d(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0)) nor = arrow3d(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0)) bin = arrow3d(pos_tzero, pos_tzero + binormal(t=t0), rgbcolor=(0,0,0.5)) acc = arrow3d(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1)) tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) ) norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1)) # # accumulate the graphic based on checkboxes # picture = path if pos_check: picture = picture + pos if vel_check: picture = picture + vel if tan_check: picture = picture+ tan if nor_check: picture = picture + nor if bin_check: picture = picture + bin if acc_check: picture = picture + acc if tancomp_check: picture = picture + tancomp if norcomp_check: picture = picture + norcomp # # print textual info # print("Position vector: r(t)=", position) print("Speed is ", N(speed(t=t0))) print("Curvature is ", N(curvature)) ## print("Torsion is ", N(torsion)) print() print("Right-click on graphic to zoom to 400%") print("Drag graphic to rotate") # # show accumulated graphical info # show(picture, aspect_ratio=[1,1,1]) 

%hide # 3-D motion and vector calculus # Copyright 2009, Robert A. Beezer # Creative Commons BY-SA 3.0 US # # # 2009/02/15 Built on Sage 3.3.rc0 # 2009/02/17 Improvements from Jason Grout # Fall 2011 Bug fixes from John Travis # # variable parameter is t # later at a particular value named t0 # # un-comment double hash (##) to get # time-consuming torsion computation # var('t') # # parameter range # start=-4*pi stop=8*pi # # position vector definition # edit here for new example # example is wide ellipse # adjust figsize in final show() to get accurate aspect ratio # a=1/(8*pi) c=(3/2)*a position=vector( (exp(a*t)*cos(t), exp(a*t)*sin(t), exp(c*t)) ) # graphic of the motion itself path = parametric_plot3d( position.list(), (t, start, stop), color = "black" ) # derivatives of motion, lengths, unit vectors, etc # velocity = derivative( position,t) acceleration = derivative(velocity,t) speed = velocity.norm() speed_deriv = derivative(speed,t) tangent = (1/speed)*velocity dT = derivative(tangent,t) normal = (1/dT.norm())*dT binormal = tangent.cross_product(normal) ## dB = derivative(binormal(t),t) # # interact section # slider for parameter, 24 settings # checkboxes for various vector displays # computations at one value of parameter, t0 # @interact def _(t0 = slider(float(start), float(stop), float((stop-start)/24), float(start) , label = "Parameter"), pos_check = ("Position", True), vel_check = ("Velocity", False), tan_check = ("Unit Tangent", False), nor_check = ("Unit Normal", False), bin_check = ("Unit Binormal", False), acc_check = ("Acceleration", True), tancomp_check = ("Tangential Component", True), norcomp_check = ("Normal Component", True) ): # # location of interest # pos_tzero = position(t=t0) # # various scalar quantities at point # speed_component = speed(t=t0) tangent_component = speed_deriv(t=t0) normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 ) curvature = normal_component/speed_component^2 ## torsion = (1/speed_component)*(dB(t=t0)).dot_product(normal(t0)) # # various vectors, mostly as arrows from the point # pos = arrow3d((0,0,0), pos_tzero, rgbcolor=(0,0,0)) tan = arrow3d(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) ) vel = arrow3d(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0)) nor = arrow3d(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0)) bin = arrow3d(pos_tzero, pos_tzero + binormal(t=t0), rgbcolor=(0,0,0.5)) acc = arrow3d(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1)) tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) ) norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1)) # # accumulate the graphic based on checkboxes # picture = path if pos_check: picture = picture + pos if vel_check: picture = picture + vel if tan_check: picture = picture+ tan if nor_check: picture = picture + nor if bin_check: picture = picture + bin if acc_check: picture = picture + acc if tancomp_check: picture = picture + tancomp if norcomp_check: picture = picture + norcomp # # print textual info # print "Position vector: r(t)=", position print "Speed is ", N(speed(t=t0)) print "Curvature is ", N(curvature) ## print "Torsion is ", N(torsion) print print "Right-click on graphic to zoom to 400%" print "Drag graphic to rotate" # # show accumulated graphical info # show(picture, aspect_ratio=[1,1,1])