#2018 NSW HSC Ext2 Maths Q13a http://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/Understanding-the-curriculum/resources/hsc-exam-papers/hsc-exam-paper-detail/2018/mathematics-extension-2-2018-hsc-exam-pack

#See https://youtu.be/rCHJefPD1js

#Usage:

#Install freecad: https://www.freecadweb.org/wiki/Download

#Paste the following into the python console (if python console is not visible, go View -> Panels -> Python Console)

import math

from FreeCAD import Base

import Part

import Draft

from PySide import QtCore

#open new document

App.newDocument("Q13a")

App.setActiveDocument("Q13a")

App.ActiveDocument=App.getDocument("Q13a")

Gui.ActiveDocument=Gui.getDocument("Q13a")

doc = FreeCAD.ActiveDocument

#show the grid

Gui.activateWorkbench("DraftWorkbench")

FreeCAD.DraftWorkingPlane.alignToPointAndAxis(FreeCAD.Vector(0,0,0), FreeCAD.Vector(0,0,1), 0)

FreeCADGui.Snapper.setGrid()

#pretty turquoise :)

face_colour = (0.0,0.85,0.75)

line_colour = (0.1,0.3,0.3)

scale = 10.0 # 1unit = 10mm

offset = 0.01 # Some things need to be offset imperceptibly from their true position to keep freecad happy

######## Axes ##########

axes = doc.addObject("App::DocumentObjectGroup","Axes")

y_end = 1.2*scale

x_start = -0.3*scale

x_end = 2.2*scale

axis_radius = 0.05

#x-axis

cyl = doc.addObject("Part::Cylinder","x-axis")

cyl.Radius = axis_radius

cyl.Height = x_end - x_start

cyl.Placement=Base.Placement(Base.Vector(x_start,0.00,0.00),App.Rotation(App.Vector(0,1,0),90))

cyl.ViewObject.DiffuseColor=[(0.0,0.0,0.0)]

axes.addObject(cyl)

txt = Draft.makeText('x', point=FreeCAD.Vector(x_end + 0.5, -0.5, 0))

txt.ViewObject.FontSize = 2

axes.addObject(txt)

#y-axis

cyl = doc.addObject("Part::Cylinder","y-axis")

cyl.Radius = axis_radius

cyl.Height = 2*y_end

cyl.Placement=Base.Placement(Base.Vector(0.00,-y_end,0.00),App.Rotation(App.Vector(1,0,0),-90))

cyl.ViewObject.DiffuseColor=[(0.0,0.0,0.0)]

axes.addObject(cyl)

txt = Draft.makeText('y', point=FreeCAD.Vector(-0.5, y_end + 1, 0))

txt.ViewObject.FontSize = 2

axes.addObject(txt)

########## Curve ############

n = 10

dx = scale/n

points = []

def my_func(x): #y^2 = x(1-x)^2

return scale * math.sqrt(x/scale*(1-x/scale)**2)

#top half (the offset is so that it doesn't try to join the first and last points with a smooth curve)

for i in range(n,0,-1):

x = i*dx

points.append(FreeCAD.Vector(x, my_func(x) + offset, 0.0))

#bottom half

for i in range(0,n+1):

x = i*dx

points.append(FreeCAD.Vector(x, -my_func(x) - offset, 0.0))

#join the points with a spline

spline = Draft.makeBSpline(points,closed=False,face=False,support=None)

#draw in the axis of rotation (x=1) as well

axis_of_rotation = Draft.makeWire([FreeCAD.Vector(scale,-scale/2,0),FreeCAD.Vector(scale,scale/2,0)],closed=False,face=False,support=None)

######## solid of revolution ####

#join the first and last points of the spline with a straight line to make a closed path

points=[FreeCAD.Vector(scale,-offset,0.0),FreeCAD.Vector(scale,offset,0.0)]

line = Draft.makeWire(points,closed=False,face=False,support=None)

line.ViewObject.Visibility=False

wire = Draft.upgrade([line,spline],delete=False)[0][0]

wire.ViewObject.Visibility=False

#Path of the sweep - arc of a circle

circle = Draft.makeCircle(radius=3.0,placement=App.Placement(App.Vector(scale - offset,0,0),App.Rotation(App.Vector(1,0,0),90)),face=False,startangle=0,endangle=360.0,support=None)

Draft.rotate([circle],180.0,FreeCAD.Vector(scale,0.0,0.0),axis=FreeCAD.Vector(0.0,0.0,1.0),copy=False) #rotate it around the z-axis 180 degrees

circle.ViewObject.Visibility=False

sweep = doc.addObject('Part::Sweep','Sweep')

sweep.Sections=[wire, ]

sweep.Spine=(circle,["Edge1"])

sweep.Solid=True

sweep.Frenet=False

sweep.ViewObject.Visibility=False

######### Animated sweep ##################

#step the angle of the arc up gradually to produce sweep animation

timer = QtCore.QTimer() #can't just use time.sleep because it blocks the display updating

angle = 0

angle_step = 5

def sweep_step():

timer.stop()

timer.timeout.connect(sweep_step)

def start_sweep():

timer.start(10)

######### Cylindrical shells ##############

shells = doc.addObject("App::DocumentObjectGroup","Shells")

def make_shell(x):

return shell

for i in range (0,n):

if i != 2: #2 is the special one which we cut and flatten

x = i*dx

shells.addObject(make_shell(x))

######### Flattened shell ################

x = 2*dx

h = my_func(x + dx/2)

#create the rectangular cross-section, which we'll sweep in an arc to create the shell

points=[FreeCAD.Vector(scale,-h,-scale + x),FreeCAD.Vector(scale,-h,-scale+x+dx),FreeCAD.Vector(scale,h,-scale + x+dx),FreeCAD.Vector(scale,h,-scale + x),FreeCAD.Vector(scale,-h,-scale + x)]

shell_wire = Draft.makeWire(points,closed=True,face=False,support=None)

shell_wire.ViewObject.Visibility = False

rotation = App.Rotation(App.Vector(1,1,-1),120) #default rotation is around the z-axis. this makes the arc go around the y-axis

arc = Draft.makeCircle(radius=3.0,placement=App.Placement(App.Vector(scale,0,0),rotation),face=False,startangle=-180,endangle=180-offset,support=None)

arc.ViewObject.Visibility = False

shell_flat = doc.addObject('Part::Sweep','Shell_flat')

shell_flat.Sections=[shell_wire, ]

shell_flat.Spine=(arc,["Edge1"])

shell_flat.Solid=True

shell_flat.Frenet=False

shells.addObject(shell_flat)

shells.ViewObject.Visibility = True

shells.ViewObject.Visibility = False

def flatten(theta): #theta = 0 is fully flat, theta = 180 is cylindrical shell

doc.recompute()

flat_timer = QtCore.QTimer()

def flatten_step(): #decrease theta

flat_timer.stop()

flat_timer.timeout.connect(flatten_step)

def start_flatten():

global angle

angle = 180

flat_timer.start(10)

############# Next view ###############

#see next.FCMacro for how to bind this to a keyboard shortcut

def next(): #step through the views, first just the graph, then the sweep, then the shells, then flatten a shell

start_sweep()