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"source": [
"In Lecture 27 of MCS 320, we make three dimensional plots."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 1. Surfaces"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Surfaces can be defined explicitly, by a function $z = f(x,y)$, or implicitly by an equation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consider the surface $z = \\cos(x y)$, for $x \\in [-\\pi, \\pi]$ and $y \\in [-\\pi, \\pi]$."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
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"execution_count": 1,
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"output_type": "execute_result"
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],
"source": [
"x, y = var('x,y')\n",
"plot3d(cos(x*y), (x, -pi, pi), (y, -pi, pi), figsize=3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can change the aspect ratio, so the plot does not look that flat."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
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"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"plot3d(cos(x*y), (x, -pi, pi), (y, -pi, pi), aspect_ratio=(1,1,2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"``[-0.2194,-0.5297,-0.8193],142.51`` is the viewpoint copied to the clipboard"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
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"text/html": [
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"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"plot3d(cos(x*y), (x, -pi, pi), (y, -pi, pi), aspect_ratio=(1,1,2)).rotateX(-0.2).rotateZ(-0.8)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
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"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"plot3d(cos(x*y), (x, -pi, pi), (y, -pi, pi), adaptive=True, \\\n",
" aspect_ratio=(1,1,2), color='automatic', num_colors=256)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot the surface defined by $z^3 - x^2 - y^2 = 0$ for $x$ and $y$ in $[-1,1]$ and for $z \\in [0,1]$."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
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"text/html": [
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"Graphics3d Object"
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"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x, y, z = var('x, y, z')\n",
"implicit_plot3d(z^3 - x^2 - y^2, (x, -1, 1), (y,-1, 1), (z, 0, 1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A surface can be defined in parameter form. As a surface is two dimensional, there are two free parameters."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
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"text/html": [
"\n",
"\n"
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"metadata": {},
"output_type": "display_data"
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],
"source": [
"u, v = var('u, v')\n",
"fx = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))\n",
"fy = sin(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))\n",
"fz = v\n",
"heart = parametric_plot3d([fx, fy, fz], (u, -pi, pi), (v, -1, 1), \\\n",
" color='red').rotateX(-pi/9).rotateY(-pi/2).rotateZ(pi/4)\n",
"show(heart, frame=False, figsize=4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 2. Space Curves"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The twisted cubic is an example of a space curve. We can define it in parameter form, or as the intersection of two cylinders."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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"source": [
"t = var('t')\n",
"twisted = (t, t^2, t^3)\n",
"pt = parametric_plot3d(twisted, (t, -1, 1), thickness=10, color='red').rotateY(pi/15).rotateZ(pi/3)\n",
"show(pt, figsize=3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The twisted cubic lies on a parabolic cylinder."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
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],
"source": [
"x, y, z = var('x,y,z')\n",
"c2 = implicit_plot3d(x^2 - y,(x,-1,1),(y,0,1), (z,-1,1)).rotateY(pi/15).rotateZ(pi/3)\n",
"(pt+c2).show(figsize=3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The twisted cubic lies in the intersection of $x^2 - y = 0$ and $x^3 - z = 0$, respectively a parabolic and a cubic cylinder."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
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