def main() -> None:
wheel1 = (
ps.Circle(ps.Point(w_1, R), R)
.set_fill_color(ps.Style.Color.BLUE)
.set_line_width(6)
)
wheel2 = wheel1.translate(ps.Point(L, 0))
under = ps.Rectangle(ps.Point(w_1 - 2 * R, 2 * R), 2 * R + L + 2 * R, H)
under.style.fill_color = ps.Style.Color.RED
under.style.line_color = ps.Style.Color.RED
over = ps.Rectangle(ps.Point(w_1, 2 * R + H), 2.5 * R, 1.25 * H).set_fill_color(
ps.Style.Color.WHITE
)
over.style.line_width = 14
over.style.line_color = ps.Style.Color.RED
over.style.fill_pattern = ps.Style.FillPattern.UP_RIGHT_TO_LEFT
ground = ps.Wall([ps.Point(w_1 - L, 0), ps.Point(w_1 + 3 * L, 0)], -0.3 * R)
ground.style.fill_pattern = ps.Style.FillPattern.UP_LEFT_TO_RIGHT
model = ps.Composition(
{
"wheel1": wheel1,
"wheel2": wheel2,
"under": under,
"over": over,
"ground": ground,
}
)
fig = ps.Figure(
0, w_1 + 2 * L + 3 * R, -1, 2 * R + 3 * H, backend=MatplotlibBackend
)
fig.add(model)
fig.show()
def main() -> None:
c = ps.Point(w_1, R)
wheel1 = ps.Circle(c, R)
wheel2 = wheel1.translate(ps.Point(L, 0))
under = ps.Rectangle(ps.Point(w_1 - 2 * R, 2 * R), 2 * R + L + 2 * R, H)
over = ps.Rectangle(ps.Point(w_1, 2 * R + H), 2.5 * R,
1.25 * H).set_fill_color(ps.Style.Color.WHITE)
ground = ps.Wall(
[ps.Point(w_1 - L, 0), ps.Point(w_1 + 3 * L, 0)], -0.3 * R)
ground.style.fill_pattern = ps.Style.FillPattern.UP_RIGHT_TO_LEFT
vehicle = ps.Composition({
"wheel1": wheel1,
"wheel2": wheel2,
"under": under,
"over": over,
"ground": ground,
})
vehicle.style.line_color = ps.Style.Color.RED
wheel1_dim = ps.LinearDimension("$w_1$", c + ps.Point(2, 0.25), c)
hdp = w_1 + L + 3 * R # horizontal dimension position
R_dim = ps.LinearDimension("$R$", ps.Point(hdp, 0), ps.Point(hdp, R))
H_dim = ps.LinearDimension("$H$", ps.Point(hdp, 2 * R),
ps.Point(hdp, 2 * R + H))
H2_dim = ps.LinearDimension("$\\frac{5}{4}H$", ps.Point(hdp, 2 * R + H),
ps.Point(hdp, 2 * R + (9 / 4) * H))
vdp = 2 * R + H + 3 / 2 * H
R2_dim = ps.LinearDimension("$2R$", ps.Point(w_1 - 2 * R, vdp),
ps.Point(w_1, vdp))
L_dim = ps.LinearDimension("$L$", ps.Point(w_1, vdp),
ps.Point(w_1 + L, vdp))
R3_dim = ps.LinearDimension("$2R$", ps.Point(w_1 + L, vdp),
ps.Point(w_1 + L + 2 * R, vdp))
dimensions = ps.Composition({
"wheel1_dim": wheel1_dim,
"R_dim": R_dim,
"H_dim": H_dim,
"H2_dim": H2_dim,
"R2_dim": R2_dim,
"L_dim": L_dim,
"R3_dim": R3_dim,
})
model = ps.Composition({"vehicle": vehicle, "dimensions": dimensions})
figure = ps.Figure(0,
w_1 + 2 * L + 3 * R,
-1,
2 * R + 3 * H,
backend=MatplotlibBackend)
figure.add(model)
figure.show()
def main() -> None:
d = make_dashpot(0)
s = make_spring(0)
M = ps.Rectangle(ps.Point(0, H), 4 * H, 4 * H).set_line_width(4)
left_wall = ps.Rectangle(ps.Point(-L, 0), H / 10, L).set_fill_pattern(
ps.Style.FillPattern.UP_LEFT_TO_RIGHT)
ground = ps.Wall([ps.Point(-L / 2, 0), ps.Point(L, 0)], thickness=-H / 10)
wheel1 = ps.Circle(ps.Point(H, H / 2), H / 2)
wheel2 = wheel1.translate(ps.Point(2 * H, 0))
fontsize = 24
text_m = ps.Text("$m$", ps.Point(2 * H, H + 2 * H))
text_m.style.font_size = fontsize
text_ku = ps.Text("$ku$", ps.Point(-L / 2, H + 4 * H))
text_ku.style.font_size = fontsize
text_bv = ps.Text("$bu'$", ps.Point(-L / 2, H))
text_bv.style.font_size = fontsize
x_axis = ps.Axis(ps.Point(2 * H, L), H, "$u(t)$")
x_axis_start = ps.Line(ps.Point(2 * H, L - H / 4),
ps.Point(2 * H, L + H / 4)).set_line_width(4)
model = ps.Composition({
"spring": s,
"mass": M,
"left wall": left_wall,
"ground": ground,
"wheel1": wheel1,
"wheel2": wheel2,
"text_m": text_m,
"text_ku": text_ku,
"x_axis": x_axis,
"x_axis_start": x_axis_start,
})
fig = ps.Figure(-L, x_max, -1, L + H, backend=MatplotlibBackend)
fig.add(model)
damping = ps.Composition({"dashpot": d, "text_bv": text_bv})
# or fig = Composition(dict(fig=fig, dashpot=d, text_bv=text_bv))
fig.add(damping)
fig.show()
P = ps.Point(W / 6, 0.85 * H) # rotation point
a = 2 * np.pi / 9 # angle
vertical = ps.Line(P, P - ps.Point(0, L))
path = ps.Arc(P, L, -np.pi / 2, a)
theta = ps.ArcWithText(r"$\theta$",
P,
L / 4,
-np.pi / 2,
a,
text_spacing=1 / 30.0)
mass_pt = path.end
rod = ps.Line(P, mass_pt)
mass = ps.Circle(mass_pt, L / 20.0)
rod_vec = rod.end - rod.start
unit_rod_vec = rod_vec.unit_vector()
mass_symbol = ps.Text("$m$", mass_pt + unit_rod_vec * (L / 10.0))
length = ps.DistanceWithText("$L$", P, mass_pt)
# Displace length indication
length = length.translate(rod_vec.normal() * (L / 15))
gravity = ps.Gravity(start=P + ps.Point(0.8 * L, 0), length=L / 3)
def set_dashed_thin_blackline(*objects: ps.Shape):
"""Set linestyle of objects to dashed, black, width=1."""
for obj in objects:
obj.set_line_style(ps.Style.LineStyle.DASHED)
def pendulum(theta, S, mg, drag) -> ps.Composition:
"""Draw a free body animation of a pendulum.
params:
theta: the angle from the vertical at which the pendulum is.
S: the force exerted toward the pivot.
mg: the force owing to gravity.
drag: the force acting against the motion of the pendulum.
return: A composition of the pendulum
"""
a = theta
P = ps.Point(W / 2, 0.9 * H) # rotation point
path = ps.Arc(P, L, -ps.Angle(np.pi / 2), a)
mass_pt = path.end
rod = ps.Line(P, mass_pt)
theta = ps.AngularDimension(r"$\theta$", P + ps.Point(0, -L / 4),
P + (mass_pt - P).unit_vector * (L / 4), P)
theta.extension_lines = False
mass = ps.Circle(mass_pt, L / 30.0).set_fill_color(ps.Style.Color.BLUE)
rod_vec = rod.end - rod.start
length = ps.LinearDimension("$L$", mass_pt, P)
# Displace length indication
length = length.translate(ps.Point(-np.cos(a), -np.sin(a)) * (L / 15.0))
length.style.line_width = 0.1
gravity_start = ps.Point(0.8 * L, 0)
gravity = ps.Gravity(P + gravity_start, L / 3)
dashed_thin_black_line = ps.Style()
dashed_thin_black_line.line_style = ps.Style.LineStyle.DASHED
dashed_thin_black_line.line_color = ps.Style.Color.BLACK
dashed_thin_black_line.line_width = 1.0
path.style = dashed_thin_black_line
vertical = ps.Line(rod.start, rod.start + ps.Point(0, -L))
vertical.style = dashed_thin_black_line
rod.style = dashed_thin_black_line
comp = ps.Composition({
"body": mass,
"rod": rod,
"vertical": vertical,
"theta": theta,
"path": path,
"g": gravity,
# "L": length,
})
magnitude = 1.2 * L / 6 # length of a unit force in figure
force = mg # constant (scaled eq: about 1)
force *= magnitude
mg_force = (ps.Force(
"$mg$",
mass_pt,
mass_pt + ps.Point(0, 1) * force,
text_position=ps.TextPosition.END,
) if force != 0 else None)
force = S
force *= magnitude
rod_force = (ps.Force(
"S",
mass_pt,
mass_pt - rod_vec.unit_vector * force,
text_position=ps.TextPosition.END,
) if force != 0 else None)
force = drag
force *= magnitude
air_force = (ps.Force(
"",
mass_pt,
mass_pt - rod_vec.normal * force,
) if force != 0 else None)
x0y0 = ps.Text("$(x_0,y_0)$", P + ps.Point(-0.4, -0.1))
ir = ps.Force(
r"$\mathbf{i}_r$",
P,
P + rod_vec.unit_vector * (L / 10),
text_position=ps.TextPosition.END,
# spacing=ps.Point(0.015, 0)
)
ith = ps.Force(
r"$\mathbf{i}_{\theta}$",
P,
P + rod_vec.normal * (L / 10),
text_position=ps.TextPosition.END,
# spacing=ps.Point(0.02, 0.005)
)
body_diagram = ps.Composition({
"mg": mg_force,
"S": rod_force,
"air": air_force,
"ir": ir,
"ith": ith,
"origin": x0y0,
})
comp = comp.merge(body_diagram)
return comp
t_max = t_mesh[-1] + 0.3 * t_axis_extent
logging.info(t_max)
u_max = 1.3 * max([u(t) for t in t_mesh])
logging.info(u_max)
u_min = -0.2 * u_max
logging.info(u_max)
r = 0.005 * (t_max - t_min) # radius of circles placed at mesh points
# import random; random.seed(12)
perturbations = [0, 0.1, 0.1, 0.2, -0.4, -0.1]
u_points = {}
u_values = []
for i, t in enumerate(t_mesh):
u_value = u(t) + perturbations[i]
u_values.append(u_value)
circle = ps.Circle(ps.Point(t, u_value),
r).set_fill_color(ps.Style.Color.BLACK)
text = ps.Text(
"$u^%d$" % i,
ps.Point(t, u_value) +
(ps.Point(0.0, 3 * r) if i > 0 else ps.Point(-3 * r, 0.0)),
)
u_points[i] = ps.Composition({"circle": circle, "u_point": text})
u_discrete = ps.Composition(u_points)
i_lines = {}
for i in range(1, len(t_mesh)):
i_lines[i] = ps.Line(ps.Point(t_mesh[i - 1], u_values[i - 1]),
ps.Point(t_mesh[i], u_values[i])).set_line_width(1)
interpolant = ps.Composition(i_lines)
x_axis_extent: float = t_mesh[-1] + 0.2 * t_axis_extent
def main() -> None:
u = ps.SketchyFunc3()
Nt = 5
t_mesh = np.linspace(0, 6, Nt + 1)
# Add 20% space to the left and 30% to the right of the coordinate system
t_axis_extent = t_mesh[-1] - t_mesh[0]
logging.info(t_axis_extent)
t_min = t_mesh[0] - 0.2 * t_axis_extent
logging.info(t_min)
t_max = t_mesh[-1] + 0.3 * t_axis_extent
logging.info(t_max)
u_max = 1.3 * max([u(t) for t in t_mesh])
logging.info(u_max)
u_min = -0.2 * u_max
logging.info(u_max)
r = 0.005 * (t_max - t_min) # radius of circles placed at mesh points
# import random; random.seed(12)
perturbations = [0, 0.1, 0.1, 0.2, -0.4, -0.1]
u_points = {}
u_values = []
for i, t in enumerate(t_mesh):
u_value = u(t) + perturbations[i]
u_values.append(u_value)
circle = ps.Circle(ps.Point(t, u_value),
r).set_fill_color(ps.Style.Color.BLACK)
text = ps.Text(
"$u^%d$" % i,
ps.Point(t, u_value) +
(ps.Point(0.0, 3 * r) if i > 0 else ps.Point(-3 * r, 0.0)),
)
u_points[i] = ps.Composition({"circle": circle, "u_point": text})
u_discrete = ps.Composition(u_points)
i_lines = {}
for i in range(1, len(t_mesh)):
i_lines[i] = ps.Line(ps.Point(t_mesh[i - 1], u_values[i - 1]),
ps.Point(t_mesh[i],
u_values[i])).set_line_width(1)
interpolant = ps.Composition(i_lines)
x_axis_extent: float = t_mesh[-1] + 0.2 * t_axis_extent
logging.info(x_axis_extent)
axes = ps.Composition({
"x":
ps.Axis(
ps.Point(0.0, 0.0),
x_axis_extent,
"$t$",
),
"y":
ps.Axis(ps.Point(0.0, 0.0),
0.8 * u_max,
"$u$",
rotation_angle=np.pi / 2),
})
h = 0.03 * u_max # tickmarks height
i_nodes = {}
for i, t in enumerate(t_mesh):
i_nodes[i] = ps.Composition({
"node":
ps.Line(ps.Point(t, h), ps.Point(t, -h)),
"name":
ps.Text("$t_%d$" % i, ps.Point(t, -3.5 * h)),
})
nodes = ps.Composition(i_nodes)
fig = ps.Figure(t_min, t_max, u_min, u_max, backend=MatplotlibBackend)
# Draw t_mesh with discrete u points
illustration = ps.Composition(dict(
u=u_discrete,
mesh=nodes,
axes=axes,
))
fig.erase()
fig.add(illustration)
fig.show()
# Add exact u line (u is a Spline Shape that applies 500 intervals by default
# for drawing the curve)
exact = u.set_line_style(ps.Style.LineStyle.DASHED).set_line_width(1)
fig.add(exact)
fig.show()
# Add linear interpolant
fig.add(interpolant)
fig.show()
# Linear interpolant without exact, smooth line
fig.erase()
fig.add(illustration)
fig.add(interpolant)
fig.show()
import pysketcher as ps
from pysketcher.backend.matplotlib import MatplotlibBackend
R = 1 # radius of wheel
L = 4 # distance between wheels
H = 2 # height of vehicle body
w_1 = 5 # position of front wheel
# TODO : draw grids
# drawing_tool.set_grid(True)
c = ps.Point(w_1, R)
wheel1 = ps.Circle(c, R)
wheel2 = wheel1.translate(ps.Point(L, 0))
under = ps.Rectangle(ps.Point(w_1 - 2 * R, 2 * R), 2 * R + L + 2 * R, H)
over = ps.Rectangle(ps.Point(w_1, 2 * R + H), 2.5 * R,
1.25 * H).set_fill_color(ps.Style.Color.WHITE)
ground = ps.Wall([ps.Point(w_1 - L, 0), ps.Point(w_1 + 3 * L, 0)], -0.3 * R)
ground.style.fill_pattern = ps.Style.FillPattern.UP_RIGHT_TO_LEFT
vehicle = ps.Composition({
"wheel1": wheel1,
"wheel2": wheel2,
"under": under,
"over": over,
"ground": ground
})
vehicle.style.line_color = ps.Style.Color.RED
def main():
u = ps.SketchyFunc3()
t_mesh = np.linspace(0, 6, Nt + 1)
t_mesh_staggered = np.linspace(0.5 * (t_mesh[0] + t_mesh[1]),
0.5 * (t_mesh[-2] + t_mesh[-1]), Nt)
# Add 20% space to the left and 30% to the right of the coordinate system
t_axis_extent = t_mesh[-1] - t_mesh[0]
t_min = t_mesh[0] - 0.2 * t_axis_extent
t_max = t_mesh[-1] + 0.3 * t_axis_extent
u_max = 1.3 * max([u(t) for t in t_mesh])
u_min = -0.2 * u_max
r = 0.005 * (t_max - t_min) # radius of circles placed at mesh Points
u_discrete = ps.Composition({
i: ps.Composition(
dict(
circle=ps.Circle(ps.Point(t, u(t)),
r).set_fill_color(ps.Style.Color.BLACK),
u_Point=ps.Text(
"$u_%d$" % i,
ps.Point(t, u(t)) +
(ps.Point(0, 5 * r) if i > 0 else ps.Point(-5 * r, 0)),
),
))
for i, t in enumerate(t_mesh)
})
# u' = v
# v = u.smooth.derivative(n=1)
v = ps.SketchyFunc4()
v_discrete = ps.Composition({
i: ps.Composition(
dict(
circle=ps.Circle(ps.Point(t, v(t)),
r).set_fill_color(ps.Style.Color.RED),
v_Point=ps.Text(
r"$v_{%d/2}$" % (2 * i + 1),
ps.Point(t, v(t)) + (ps.Point(0, 5 * r)),
),
))
for i, t in enumerate(t_mesh_staggered)
})
axes = ps.Composition(
dict(
x=ps.Axis(ps.Point(0, 0), t_mesh[-1] + 0.2 * t_axis_extent, "$t$"),
y=ps.Axis(ps.Point(0, 0),
0.8 * u_max,
"$u,v$",
rotation_angle=np.pi / 2),
))
h = 0.03 * u_max # tickmarks height
u_nodes = ps.Composition({
i: ps.Composition(
dict(
node=ps.Line(ps.Point(t, h), ps.Point(t, -h)),
name=ps.Text("$t_%d$" % i, ps.Point(t, -3.5 * h)),
))
for i, t in enumerate(t_mesh)
})
v_nodes = ps.Composition({
i: ps.Composition(
dict(
node=ps.Line(ps.Point(t, h / 1.5), ps.Point(
t, -h / 1.5)).set_line_color(ps.Style.Color.RED),
name=ps.Text(r"$t_{%d/2}$" % (2 * i + 1),
ps.Point(t, -3.5 * h)),
))
for i, t in enumerate(t_mesh_staggered)
})
illustration = ps.Composition(
dict(u=u_discrete,
v=v_discrete,
u_mesh=u_nodes,
v_mesh=v_nodes,
axes=axes))
fig = ps.Figure(t_min, t_max, u_min, u_max, backend=MatplotlibBackend)
# Staggered t mesh and u and v Points
fig.add(illustration)
fig.show()
# Exact u line (u is a Spline Shape that applies 500 intervals by default
# for drawing the curve)
u_exact = u.set_line_style(ps.Style.LineStyle.DASHED).set_line_width(1)
fig.add(u_exact)
fig.show()
# v = Curve(u.xcoor, v(u.xcoor))
t_mesh_staggered_fine = np.linspace(t_mesh_staggered[0],
t_mesh_staggered[-1], 501)
t_mesh_staggered_points = [
ps.Point(x, v(x)) for x in t_mesh_staggered_fine
]
v_exact = (ps.Curve(t_mesh_staggered_points).set_line_style(
ps.Style.LineStyle.DASHED).set_line_width(1))
fig.add(v_exact)
fig.show()
def make_spring(x):
s_start = ps.Point(-L, 4 * H)
s = ps.Spring(start=s_start, length=L + x, bar_length=3 * H / 2)
s = s.rotate(-np.pi / 2, s_start)
return s
d = make_dashpot(0)
s = make_spring(0)
M = ps.Rectangle(ps.Point(0, H), 4 * H, 4 * H).set_line_width(4)
left_wall = ps.Rectangle(ps.Point(-L, 0), H / 10, L).set_fill_pattern(
ps.Style.FillPattern.UP_LEFT_TO_RIGHT
)
ground = ps.Wall([ps.Point(-L / 2, 0), ps.Point(L, 0)], thickness=-H / 10)
wheel1 = ps.Circle(ps.Point(H, H / 2), H / 2)
wheel2 = wheel1.translate(ps.Point(2 * H, 0))
fontsize = 24
text_m = ps.Text("$m$", ps.Point(2 * H, H + 2 * H))
text_m.style.font_size = fontsize
text_ku = ps.Text("$ku$", ps.Point(-L / 2, H + 4 * H))
text_ku.style.font_size = fontsize
text_bv = ps.Text("$bu'$", ps.Point(-L / 2, H))
text_bv.style.font_size = fontsize
x_axis = ps.Axis(ps.Point(2 * H, L), H, "$u(t)$", label_spacing=(0.04, -0.01))
x_axis_start = ps.Line(
ps.Point(2 * H, L - H / 4), ps.Point(2 * H, L + H / 4)
).set_line_width(4)
model = ps.Composition(
t_mesh = np.linspace(0, 6, Nt + 1)
t_mesh_staggered = np.linspace(0.5 * (t_mesh[0] + t_mesh[1]),
0.5 * (t_mesh[-2] + t_mesh[-1]), Nt)
# Add 20% space to the left and 30% to the right of the coordinate system
t_axis_extent = t_mesh[-1] - t_mesh[0]
t_min = t_mesh[0] - 0.2 * t_axis_extent
t_max = t_mesh[-1] + 0.3 * t_axis_extent
u_max = 1.3 * max([u(t) for t in t_mesh])
u_min = -0.2 * u_max
r = 0.005 * (t_max - t_min) # radius of circles placed at mesh Points
u_discrete = ps.Composition({
i: ps.Composition(
dict(
circle=ps.Circle(ps.Point(t, u(t)),
r).set_fill_color(ps.Style.Color.BLACK),
u_Point=ps.Text(
"$u_%d$" % i,
ps.Point(t, u(t)) +
(ps.Point(0, 5 * r) if i > 0 else ps.Point(-5 * r, 0)),
),
))
for i, t in enumerate(t_mesh)
})
# u' = v
# v = u.smooth.derivative(n=1)
v = ps.SketchyFunc4()
v_discrete = ps.Composition({
i: ps.Composition(
"""Illustrate forward, backward and centered finite differences in four figures."""
import pysketcher as ps
from pysketcher.backend.matplotlib import MatplotlibBackend
xaxis = 2
f = ps.SketchyFunc1("$u(t)$")
x = 3 # center point where we want the derivative
xb = 2 # x point used for backward difference
xf = 4 # x point used for forward difference
p = ps.Point(x, f(x)) # center point
pf = ps.Point(xf, f(xf)) # forward point
pb = ps.Point(xb, f(xb)) # backward point
r = 0.1 # radius of circles placed at key points
c = ps.Circle(p, r).set_line_color(ps.Style.Color.BLUE)
cf = ps.Circle(pf, r).set_line_color(ps.Style.Color.RED)
cb = ps.Circle(pb, r).set_line_color(ps.Style.Color.GREEN)
# Points in the mesh
p0 = ps.Point(x, xaxis) # center point
pf0 = ps.Point(xf, xaxis) # forward point
pb0 = ps.Point(xb, xaxis) # backward point
tick = 0.05
axis = ps.Composition(
{
"hline": ps.Line(pf0 - ps.Point(3, 0), pb0 + ps.Point(3, 0))
.set_line_color(ps.Style.Color.BLACK)
.set_line_width(1),
"tick_m1": ps.Line(pf0 + ps.Point(0, tick), pf0 - ps.Point(0, tick))
import pysketcher as ps
from pysketcher.backend.matplotlib import MatplotlibBackend
R = 1 # radius of wheel
L = 4 # distance between wheels
H = 2 # height of vehicle body
w_1 = 5 # position of front wheel
wheel1 = (ps.Circle(ps.Point(w_1, R),
R).set_fill_color(ps.Style.Color.BLUE).set_line_width(6))
wheel2 = wheel1.translate(ps.Point(L, 0))
under = ps.Rectangle(ps.Point(w_1 - 2 * R, 2 * R), 2 * R + L + 2 * R, H)
under.style.fill_color = ps.Style.Color.RED
under.style.line_color = ps.Style.Color.RED
over = ps.Rectangle(ps.Point(w_1, 2 * R + H), 2.5 * R,
1.25 * H).set_fill_color(ps.Style.Color.WHITE)
over.style.line_width = 14
over.style.line_color = ps.Style.Color.RED
over.style.fill_pattern = ps.Style.FillPattern.UP_RIGHT_TO_LEFT
ground = ps.Wall([ps.Point(w_1 - L, 0), ps.Point(w_1 + 3 * L, 0)], -0.3 * R)
ground.style.fill_pattern = ps.Style.FillPattern.UP_LEFT_TO_RIGHT
model = ps.Composition({
"wheel1": wheel1,
"wheel2": wheel2,
"under": under,
"over": over,
"ground": ground
})
