{
  "index": "1951-B-5",
  "type": "GEO",
  "tag": [
    "GEO",
    "ANA"
  ],
  "difficulty": "",
  "question": "5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.",
  "solution": "Solution. We can choose a coordinate system so that the origin is at the center of the torus and the \\( z \\)-axis is the axis of rotational symmetry. Then the equation of the torus is\n\\[\n\\left[\\sqrt{\\left(x^{2}+\\sqrt[y^{2}]{ }\\right)}-a\\right]^{2}+z^{2}=b^{2}\n\\]\nwhere \\( a>b>0 \\). Let \\( \\Pi \\) be a plane tangent to the torus at a point \\( \\boldsymbol{Q} \\). Rotate the coordinate system about the \\( z \\)-axis so that \\( Q \\) lies in the \\( x z \\)-plane. Now \\( \\Pi \\) intersects the \\( x z \\)-plane in a line \\( z=\\lambda x \\). We can suppose, without loss of generality, that \\( \\lambda>0 \\).\n\nThe line through \\( \\boldsymbol{Q} \\) normal to the \\( x z \\)-plane is tangent to the torus and therefore lies in \\( \\Pi \\). Hence the equation of \\( \\Pi \\) is \\( z=\\lambda x \\).\n\nIt is clear from symmetry that \\( \\Pi \\) is also tangent to the torus at a second point \\( Q^{\\prime} \\). Let \\( c \\) denote the distance from \\( O \\) to \\( Q \\). Then \\( c^{2}+b^{2}=a^{2} \\) and \\( \\lambda=b / c \\).\n\nThe intersection set \\( I \\) of \\( \\Pi \\) and the torus certainly contains the following six points: four points of the form ( \\( 0, \\pm a \\pm b, 0 \\) ) on the \\( y \\) axis, and the two points \\( Q \\) and \\( Q^{\\prime} \\). We note that the set \\( I \\) is also symmetric in the \\( y \\)-axis. Hence if \\( I \\) consists of two circles, they must be the circles of radius \\( a \\) and centers ( \\( 0, \\pm b, 0 \\) ) lying in the plane \\( \\Pi \\). (See figure.)\n\nParametric equations for the first of these circles are\n\\[\n\\begin{array}{l}\ny=b+a \\sin \\theta \\\\\nx=c \\cos \\theta \\\\\nz=b \\cos \\theta\n\\end{array}\n\\]\n\nTo check this, note that these values satisfy both \\( x^{2}+(y-b)^{2}+z^{2}=a^{2} \\) and \\( z=\\lambda x \\), so (1) describes a closed curve that lies on the sphere of radius \\( a \\) about \\( (0, b, 0) \\) and on the plane \\( z=\\lambda x \\). (To get the second circle just reverse the sign of \\( b \\).)\n\nNext we show that these circles lie on the torus. For any point on (1) we have\n\\[\n\\begin{aligned}\nx^{2}+y^{2} & =c^{2} \\cos ^{2} \\theta+b^{2}+2 a b \\sin \\theta+a^{2} \\sin ^{2} \\theta \\\\\n& =\\left(a^{2}-b^{2}\\right) \\cos ^{2} \\theta+b^{2}+2 a b \\sin \\theta+a^{2} \\sin ^{2} \\theta \\\\\n& =a^{2}+2 a b \\sin \\theta+b^{2} \\sin ^{2} \\theta=(a+b \\sin \\theta)^{2}\n\\end{aligned}\n\\]\n\nSince \\( a>b, a+b \\sin \\theta>0 \\); so \\( \\sqrt{x^{2}}+y^{2}=a+b \\sin \\theta \\). Hence\n\\[\n\\left[\\sqrt{x^{2}}+y^{2}-a\\right]^{2}+z^{2}=b^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta=b^{2}\n\\]\nwhich is the defining equation of the torus.\nThus the first circle lies on the torus. That the second circle does also follows by similar algebra, or by noting that the two circles are interchanged by a reflection in the \\( x z \\)-plane.\n\nRemark. This solution is not quite complete, because it only shows that \\( I \\) contains the two circles; conceivably there are further points in the intersection set \\( I \\), although it is intuitively evident that there are not.\n\nA complete solution can be obtained by purely algebraic manipulations. If the radical is eliminated from the original equation for the torus, we obtain the equivalent equation\n\\[\n\\left(x^{2}+y^{2}+z^{2}+a^{2}-b^{2}\\right)^{2}=4 a^{2} x^{2}+4 a^{2} y^{2}\n\\]\n\nThis can be rewritten as\n\\[\n\\left(x^{2}+y^{2}+z^{2}+b^{2}-a^{2}\\right)^{2}-4 b^{2} y^{2}=4 b^{2} x^{2}-4 c^{2} z^{2}\n\\]\nand then as\n\\[\n\\begin{aligned}\n\\left(x^{2}+(y-b)^{2}+z^{2}-a^{2}\\right)\\left(x^{2}+(y+b)^{2}+\\right. & \\left.z^{2}-a^{2}\\right) \\\\\n& =4(b x-c z)(b x+c z)\n\\end{aligned}\n\\]\n\nFrom this equation it appears that any point on both the torus and the plane \\( b x-c z=0 \\) is also on one of the two spheres\n\\[\nx^{2}+(y-b)^{2}+z^{2}=a^{2} \\text { and } x^{2}+(y+b)^{2}+z^{2}=a^{2}\n\\]\n\nConversely, any point on the plane and either sphere lies also on the torus. Thus the intersection set \\( I \\) is exactly the union of the two circles.\n\nRemark. This problem is discussed in H. S. M. Coxeter, Introduction to Geometry, Wiley, New York, 1961, pages 132-133. References to other papers may be found in Coxeter's book.",
  "vars": [
    "x",
    "y",
    "z",
    "\\\\theta"
  ],
  "params": [
    "a",
    "b",
    "c",
    "\\\\lambda",
    "O",
    "Q",
    "I"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "horizntal",
        "y": "lateralv",
        "z": "vertical",
        "\\theta": "angleth",
        "a": "majorrad",
        "b": "minorrad",
        "c": "distanc",
        "\\lambda": "planefac",
        "O": "originpt",
        "Q": "tangentp",
        "I": "interset"
      },
      "question": "5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.",
      "solution": "Solution. We can choose a coordinate system so that the origin is at the center of the torus and the \\( vertical \\)-axis is the axis of rotational symmetry. Then the equation of the torus is\n\\[\n\\left[\\sqrt{\\left(horizntal^{2}+\\sqrt[lateralv^{2}]{ }\\right)}-majorrad\\right]^{2}+vertical^{2}=minorrad^{2}\n\\]\nwhere \\( majorrad>minorrad>0 \\). Let \\( \\Pi \\) be a plane tangent to the torus at a point \\( \\boldsymbol{tangentp} \\). Rotate the coordinate system about the \\( vertical \\)-axis so that \\( tangentp \\) lies in the \\( horizntal\\,vertical \\)-plane. Now \\( \\Pi \\) intersects the \\( horizntal\\,vertical \\)-plane in a line \\( vertical=planefac\\,horizntal \\). We can suppose, without loss of generality, that \\( planefac>0 \\).\n\nThe line through \\( \\boldsymbol{tangentp} \\) normal to the \\( horizntal\\,vertical \\)-plane is tangent to the torus and therefore lies in \\( \\Pi \\). Hence the equation of \\( \\Pi \\) is \\( vertical=planefac\\,horizntal \\).\n\nIt is clear from symmetry that \\( \\Pi \\) is also tangent to the torus at a second point \\( tangentp^{\\prime} \\). Let \\( distanc \\) denote the distance from \\( originpt \\) to \\( tangentp \\). Then \\( distanc^{2}+minorrad^{2}=majorrad^{2} \\) and \\( planefac=minorrad/distanc \\).\n\nThe intersection set \\( interset \\) of \\( \\Pi \\) and the torus certainly contains the following six points: four points of the form ( \\( 0,\\pm majorrad\\pm minorrad,0 \\) ) on the \\( lateralv \\)-axis, and the two points \\( tangentp \\) and \\( tangentp^{\\prime} \\). We note that the set \\( interset \\) is also symmetric in the \\( lateralv \\)-axis. Hence if \\( interset \\) consists of two circles, they must be the circles of radius \\( majorrad \\) and centers ( \\( 0,\\pm minorrad,0 \\) ) lying in the plane \\( \\Pi \\). (See figure.)\n\nParametric equations for the first of these circles are\n\\[\n\\begin{array}{l}\nlateralv=minorrad+majorrad\\sin angleth \\\\\nhorizntal=distanc\\cos angleth \\\\\nvertical=minorrad\\cos angleth\n\\end{array}\n\\]\n\nTo check this, note that these values satisfy both \\( horizntal^{2}+(lateralv-minorrad)^{2}+vertical^{2}=majorrad^{2} \\) and \\( vertical=planefac\\,horizntal \\), so (1) describes a closed curve that lies on the sphere of radius \\( majorrad \\) about \\( (0,minorrad,0) \\) and on the plane \\( vertical=planefac\\,horizntal \\). (To get the second circle just reverse the sign of \\( minorrad \\).)\n\nNext we show that these circles lie on the torus. For any point on (1) we have\n\\[\n\\begin{aligned}\nhorizntal^{2}+lateralv^{2}&=distanc^{2}\\cos^{2}angleth+minorrad^{2}+2\\,majorrad\\,minorrad\\sin angleth+majorrad^{2}\\sin^{2}angleth\\\\&=\\left(majorrad^{2}-minorrad^{2}\\right)\\cos^{2}angleth+minorrad^{2}+2\\,majorrad\\,minorrad\\sin angleth+majorrad^{2}\\sin^{2}angleth\\\\&=majorrad^{2}+2\\,majorrad\\,minorrad\\sin angleth+minorrad^{2}\\sin^{2}angleth=(majorrad+minorrad\\sin angleth)^{2}\n\\end{aligned}\n\\]\n\nSince \\( majorrad>minorrad,\\,majorrad+minorrad\\sin angleth>0 \\); so \\( \\sqrt{horizntal^{2}+lateralv^{2}}=majorrad+minorrad\\sin angleth \\). Hence\n\\[\n\\left[\\sqrt{horizntal^{2}+lateralv^{2}}-majorrad\\right]^{2}+vertical^{2}=minorrad^{2}\\sin^{2}angleth+minorrad^{2}\\cos^{2}angleth=minorrad^{2}\n\\]\nwhich is the defining equation of the torus. Thus the first circle lies on the torus. That the second circle does also follows by similar algebra, or by noting that the two circles are interchanged by a reflection in the \\( horizntal\\,vertical \\)-plane.\n\nRemark. This solution is not quite complete, because it only shows that \\( interset \\) contains the two circles; conceivably there are further points in the intersection set \\( interset \\), although it is intuitively evident that there are not.\n\nA complete solution can be obtained by purely algebraic manipulations. If the radical is eliminated from the original equation for the torus, we obtain the equivalent equation\n\\[\n\\left(horizntal^{2}+lateralv^{2}+vertical^{2}+majorrad^{2}-minorrad^{2}\\right)^{2}=4\\,majorrad^{2}\\,horizntal^{2}+4\\,majorrad^{2}\\,lateralv^{2}\n\\]\n\nThis can be rewritten as\n\\[\n\\left(horizntal^{2}+lateralv^{2}+vertical^{2}+minorrad^{2}-majorrad^{2}\\right)^{2}-4\\,minorrad^{2}\\,lateralv^{2}=4\\,minorrad^{2}\\,horizntal^{2}-4\\,distanc^{2}\\,vertical^{2}\n\\]\nand then as\n\\[\n\\begin{aligned}\n\\left(horizntal^{2}+(lateralv-minorrad)^{2}+vertical^{2}-majorrad^{2}\\right)\\left(horizntal^{2}+(lateralv+minorrad)^{2}+vertical^{2}-majorrad^{2}\\right)\\\\=4(minorrad\\,horizntal-distanc\\,vertical)(minorrad\\,horizntal+distanc\\,vertical)\n\\end{aligned}\n\\]\n\nFrom this equation it appears that any point on both the torus and the plane \\( minorrad\\,horizntal-distanc\\,vertical=0 \\) is also on one of the two spheres\n\\[\n horizntal^{2}+(lateralv-minorrad)^{2}+vertical^{2}=majorrad^{2}\\quad\\text{and}\\quad horizntal^{2}+(lateralv+minorrad)^{2}+vertical^{2}=majorrad^{2}\n\\]\n\nConversely, any point on the plane and either sphere lies also on the torus. Thus the intersection set \\( interset \\) is exactly the union of the two circles.\n\nRemark. This problem is discussed in H. S. M. Coxeter, Introduction to Geometry, Wiley, New York, 1961, pages 132-133. References to other papers may be found in Coxeter's book."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "grapefruit",
        "y": "lemonade",
        "z": "honeycomb",
        "\\theta": "doorknobs",
        "a": "seashells",
        "b": "drumstick",
        "c": "knapsack",
        "\\lambda": "saffroned",
        "O": "flagstone",
        "Q": "rainstorm",
        "I": "playhouse"
      },
      "question": "5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.",
      "solution": "Solution. We can choose a coordinate system so that the origin is at the center of the torus and the \\( honeycomb \\)-axis is the axis of rotational symmetry. Then the equation of the torus is\n\\[\n\\left[\\sqrt{\\left(grapefruit^{2}+\\sqrt[lemonade^{2}]{ }\\right)}-seashells\\right]^{2}+honeycomb^{2}=drumstick^{2}\n\\]\nwhere \\( seashells>drumstick>0 \\). Let \\( \\Pi \\) be a plane tangent to the torus at a point \\( \\boldsymbol{rainstorm} \\). Rotate the coordinate system about the \\( honeycomb \\)-axis so that \\( rainstorm \\) lies in the \\( grapefruit honeycomb \\)-plane. Now \\( \\Pi \\) intersects the \\( grapefruit honeycomb \\)-plane in a line \\( honeycomb=saffroned grapefruit \\). We can suppose, without loss of generality, that \\( saffroned>0 \\).\n\nThe line through \\( \\boldsymbol{rainstorm} \\) normal to the \\( grapefruit honeycomb \\)-plane is tangent to the torus and therefore lies in \\( \\Pi \\). Hence the equation of \\( \\Pi \\) is \\( honeycomb=saffroned grapefruit \\).\n\nIt is clear from symmetry that \\( \\Pi \\) is also tangent to the torus at a second point \\( rainstorm^{\\prime} \\). Let \\( knapsack \\) denote the distance from \\( flagstone \\) to \\( rainstorm \\). Then \\( knapsack^{2}+drumstick^{2}=seashells^{2} \\) and \\( saffroned=drumstick / knapsack \\).\n\nThe intersection set \\( playhouse \\) of \\( \\Pi \\) and the torus certainly contains the following six points: four points of the form ( \\( 0, \\pm seashells \\pm drumstick, 0 \\) ) on the \\( lemonade \\) axis, and the two points \\( rainstorm \\) and \\( rainstorm^{\\prime} \\). We note that the set \\( playhouse \\) is also symmetric in the \\( lemonade \\)-axis. Hence if \\( playhouse \\) consists of two circles, they must be the circles of radius \\( seashells \\) and centers ( \\( 0, \\pm drumstick, 0 \\) ) lying in the plane \\( \\Pi \\). (See figure.)\n\nParametric equations for the first of these circles are\n\\[\n\\begin{array}{l}\nlemonade=drumstick+seashells \\sin doorknobs \\\\\ngrapefruit=knapsack \\cos doorknobs \\\\\nhoneycomb=drumstick \\cos doorknobs\n\\end{array}\n\\]\n\nTo check this, note that these values satisfy both \\( grapefruit^{2}+(lemonade-drumstick)^{2}+honeycomb^{2}=seashells^{2} \\) and \\( honeycomb=saffroned grapefruit \\), so (1) describes a closed curve that lies on the sphere of radius \\( seashells \\) about \\( (0, drumstick, 0) \\) and on the plane \\( honeycomb=saffroned grapefruit \\). (To get the second circle just reverse the sign of \\( drumstick \\).)\n\nNext we show that these circles lie on the torus. For any point on (1) we have\n\\[\n\\begin{aligned}\ngrapefruit^{2}+lemonade^{2} & =knapsack^{2} \\cos ^{2} doorknobs+drumstick^{2}+2 seashells drumstick \\sin doorknobs+seashells^{2} \\sin ^{2} doorknobs \\\\\n& =(seashells^{2}-drumstick^{2}) \\cos ^{2} doorknobs+drumstick^{2}+2 seashells drumstick \\sin doorknobs+seashells^{2} \\sin ^{2} doorknobs \\\\\n& =seashells^{2}+2 seashells drumstick \\sin doorknobs+drumstick^{2} \\sin ^{2} doorknobs=(seashells+drumstick \\sin doorknobs)^{2}\n\\end{aligned}\n\\]\n\nSince \\( seashells>drumstick, seashells+drumstick \\sin doorknobs>0 \\); so \\( \\sqrt{grapefruit^{2}}+lemonade^{2}=seashells+drumstick \\sin doorknobs \\). Hence\n\\[\n\\left[\\sqrt{grapefruit^{2}}+lemonade^{2}-seashells\\right]^{2}+honeycomb^{2}=drumstick^{2} \\sin ^{2} doorknobs+drumstick^{2} \\cos ^{2} doorknobs=drumstick^{2}\n\\]\nwhich is the defining equation of the torus.\nThus the first circle lies on the torus. That the second circle does also follows by similar algebra, or by noting that the two circles are interchanged by a reflection in the \\( grapefruit honeycomb \\)-plane.\n\nRemark. This solution is not quite complete, because it only shows that \\( playhouse \\) contains the two circles; conceivably there are further points in the intersection set \\( playhouse \\), although it is intuitively evident that there are not.\n\nA complete solution can be obtained by purely algebraic manipulations. If the radical is eliminated from the original equation for the torus, we obtain the equivalent equation\n\\[\n\\left(grapefruit^{2}+lemonade^{2}+honeycomb^{2}+seashells^{2}-drumstick^{2}\\right)^{2}=4 seashells^{2} grapefruit^{2}+4 seashells^{2} lemonade^{2}\n\\]\n\nThis can be rewritten as\n\\[\n\\left(grapefruit^{2}+lemonade^{2}+honeycomb^{2}+drumstick^{2}-seashells^{2}\\right)^{2}-4 drumstick^{2} lemonade^{2}=4 drumstick^{2} grapefruit^{2}-4 knapsack^{2} honeycomb^{2}\n\\]\nand then as\n\\[\n\\begin{aligned}\n\\left(grapefruit^{2}+(lemonade-drumstick)^{2}+honeycomb^{2}-seashells^{2}\\right)\\left(grapefruit^{2}+(lemonade+drumstick)^{2}+\\right. & \\left.honeycomb^{2}-seashells^{2}\\right) \\\\\n& =4(drumstick grapefruit-knapsack honeycomb)(drumstick grapefruit+knapsack honeycomb)\n\\end{aligned}\n\\]\n\nFrom this equation it appears that any point on both the torus and the plane \\( drumstick grapefruit-knapsack honeycomb=0 \\) is also on one of the two spheres\n\\[\ngrapefruit^{2}+(lemonade-drumstick)^{2}+honeycomb^{2}=seashells^{2} \\text { and } grapefruit^{2}+(lemonade+drumstick)^{2}+honeycomb^{2}=seashells^{2}\n\\]\n\nConversely, any point on the plane and either sphere lies also on the torus. Thus the intersection set \\( playhouse \\) is exactly the union of the two circles.\n\nRemark. This problem is discussed in H. S. M. Coxeter, Introduction to Geometry, Wiley, New York, 1961, pages 132-133. References to other papers may be found in Coxeter's book."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "stationary",
        "y": "flatline",
        "z": "grounded",
        "\\theta": "straighten",
        "a": "smallscale",
        "b": "largeradius",
        "c": "proximity",
        "\\lambda": "flatness",
        "O": "infinity",
        "Q": "nowherept",
        "I": "disjointset"
      },
      "question": "5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.",
      "solution": "Solution. We can choose a coordinate system so that the origin is at the center of the torus and the \\( grounded \\)-axis is the axis of rotational symmetry. Then the equation of the torus is\n\\[\n\\left[\\sqrt{\\left(stationary^{2}+\\sqrt[flatline^{2}]{ }\\right)}-smallscale\\right]^{2}+grounded^{2}=largeradius^{2}\n\\]\nwhere \\( smallscale>largeradius>0 \\). Let \\( \\Pi \\) be a plane tangent to the torus at a point \\( \\boldsymbol{nowherept} \\). Rotate the coordinate system about the \\( grounded \\)-axis so that \\( nowherept \\) lies in the \\( stationary grounded \\)-plane. Now \\( \\Pi \\) intersects the \\( stationary grounded \\)-plane in a line \\( grounded=flatness\\,stationary \\). We can suppose, without loss of generality, that \\( flatness>0 \\).\n\nThe line through \\( \\boldsymbol{nowherept} \\) normal to the \\( stationary grounded \\)-plane is tangent to the torus and therefore lies in \\( \\Pi \\). Hence the equation of \\( \\Pi \\) is \\( grounded=flatness\\,stationary \\).\n\nIt is clear from symmetry that \\( \\Pi \\) is also tangent to the torus at a second point \\( nowherept^{\\prime} \\). Let \\( proximity \\) denote the distance from \\( infinity \\) to \\( nowherept \\). Then \\( proximity^{2}+largeradius^{2}=smallscale^{2} \\) and \\( flatness=largeradius / proximity \\).\n\nThe intersection set \\( disjointset \\) of \\( \\Pi \\) and the torus certainly contains the following six points: four points of the form ( \\( 0, \\pm smallscale \\pm largeradius, 0 \\) ) on the \\( flatline \\) axis, and the two points \\( nowherept \\) and \\( nowherept^{\\prime} \\). We note that the set \\( disjointset \\) is also symmetric in the \\( flatline \\)-axis. Hence if \\( disjointset \\) consists of two circles, they must be the circles of radius \\( smallscale \\) and centers ( \\( 0, \\pm largeradius, 0 \\) ) lying in the plane \\( \\Pi \\). (See figure.)\n\nParametric equations for the first of these circles are\n\\[\n\\begin{array}{l}\nflatline=largeradius+smallscale \\sin straighten \\\\\nstationary=proximity \\cos straighten \\\\\ngrounded=largeradius \\cos straighten\n\\end{array}\n\\]\n\nTo check this, note that these values satisfy both \\( stationary^{2}+(flatline-largeradius)^{2}+grounded^{2}=smallscale^{2} \\) and \\( grounded=flatness\\,stationary \\), so (1) describes a closed curve that lies on the sphere of radius \\( smallscale \\) about \\( (0, largeradius, 0) \\) and on the plane \\( grounded=flatness\\,stationary \\). (To get the second circle just reverse the sign of \\( largeradius \\).)\n\nNext we show that these circles lie on the torus. For any point on (1) we have\n\\[\n\\begin{aligned}\nstationary^{2}+flatline^{2} &=proximity^{2} \\cos ^{2} straighten+largeradius^{2}+2 smallscale\\,largeradius \\sin straighten+smallscale^{2} \\sin ^{2} straighten \\\\\n&=(smallscale^{2}-largeradius^{2}) \\cos ^{2} straighten+largeradius^{2}+2 smallscale\\,largeradius \\sin straighten+smallscale^{2} \\sin ^{2} straighten \\\\\n&=smallscale^{2}+2 smallscale\\,largeradius \\sin straighten+largeradius^{2} \\sin ^{2} straighten=(smallscale+largeradius \\sin straighten)^{2}\n\\end{aligned}\n\\]\n\nSince \\( smallscale>largeradius,\\;smallscale+largeradius \\sin straighten>0 \\); so \\( \\sqrt{stationary^{2}}+flatline^{2}=smallscale+largeradius \\sin straighten \\). Hence\n\\[\n\\left[\\sqrt{stationary^{2}}+flatline^{2}-smallscale\\right]^{2}+grounded^{2}=largeradius^{2} \\sin ^{2} straighten+largeradius^{2} \\cos ^{2} straighten=largeradius^{2}\n\\]\nwhich is the defining equation of the torus.\nThus the first circle lies on the torus. That the second circle does also follows by similar algebra, or by noting that the two circles are interchanged by a reflection in the \\( stationary grounded \\)-plane.\n\nRemark. This solution is not quite complete, because it only shows that \\( disjointset \\) contains the two circles; conceivably there are further points in the intersection set \\( disjointset \\), although it is intuitively evident that there are not.\n\nA complete solution can be obtained by purely algebraic manipulations. If the radical is eliminated from the original equation for the torus, we obtain the equivalent equation\n\\[\n\\left(stationary^{2}+flatline^{2}+grounded^{2}+smallscale^{2}-largeradius^{2}\\right)^{2}=4 smallscale^{2} stationarity^{2}+4 smallscale^{2} flatline^{2}\n\\]\n\nThis can be rewritten as\n\\[\n\\left(stationary^{2}+flatline^{2}+grounded^{2}+largeradius^{2}-smallscale^{2}\\right)^{2}-4 largeradius^{2} flatline^{2}=4 largeradius^{2} stationary^{2}-4 proximity^{2} grounded^{2}\n\\]\n\nand then as\n\\[\n\\begin{aligned}\n\\left(stationary^{2}+(flatline-largeradius)^{2}+grounded^{2}-smallscale^{2}\\right)\\left(stationary^{2}+(flatline+largeradius)^{2}+\\right.&\\left.grounded^{2}-smallscale^{2}\\right) \\\\\n&=4(largeradius\\,stationary-proximity\\,grounded)(largeradius\\,stationary+proximity\\,grounded)\n\\end{aligned}\n\\]\n\nFrom this equation it appears that any point on both the torus and the plane \\( largeradius\\,stationary-proximity\\,grounded=0 \\) is also on one of the two spheres\n\\[\nstationary^{2}+(flatline-largeradius)^{2}+grounded^{2}=smallscale^{2} \\text{ and } stationary^{2}+(flatline+largeradius)^{2}+grounded^{2}=smallscale^{2}\n\\]\n\nConversely, any point on the plane and either sphere lies also on the torus. Thus the intersection set \\( disjointset \\) is exactly the union of the two circles.\n\nRemark. This problem is discussed in H. S. M. Coxeter, Introduction to Geometry, Wiley, New York, 1961, pages 132–133. References to other papers may be found in Coxeter's book."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "z": "kovmlpqa",
        "\\theta": "frewpdsk",
        "a": "vutnqrsa",
        "b": "gpldxeio",
        "c": "jsmbwzoh",
        "\\lambda": "rypqnesh",
        "O": "xurqpleta",
        "Q": "pzomcstn",
        "I": "vhgalezr"
      },
      "question": "5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.",
      "solution": "Solution. We can choose a coordinate system so that the origin is at the center of the torus and the \\( kovmlpqa \\)-axis is the axis of rotational symmetry. Then the equation of the torus is\n\\[\n\\left[\\sqrt{\\left(qzxwvtnp^{2}+\\sqrt[hjgrksla^{2}]{ }\\right)}-vutnqrsa\\right]^{2}+kovmlpqa^{2}=gpldxeio^{2}\n\\]\nwhere \\( vutnqrsa>gpldxeio>0 \\). Let \\( \\Pi \\) be a plane tangent to the torus at a point \\( \\boldsymbol{pzomcstn} \\). Rotate the coordinate system about the \\( kovmlpqa \\)-axis so that \\( pzomcstn \\) lies in the \\( qzxwvtnp\\,kovmlpqa \\)-plane. Now \\( \\Pi \\) intersects the \\( qzxwvtnp\\,kovmlpqa \\)-plane in a line \\( kovmlpqa=rypqnesh\\,qzxwvtnp \\). We can suppose, without loss of generality, that \\( rypqnesh>0 \\).\n\nThe line through \\( \\boldsymbol{pzomcstn} \\) normal to the \\( qzxwvtnp\\,kovmlpqa \\)-plane is tangent to the torus and therefore lies in \\( \\Pi \\). Hence the equation of \\( \\Pi \\) is \\( kovmlpqa=rypqnesh\\,qzxwvtnp \\).\n\nIt is clear from symmetry that \\( \\Pi \\) is also tangent to the torus at a second point \\( pzomcstn^{\\prime} \\). Let \\( jsmbwzoh \\) denote the distance from \\( xurqpleta \\) to \\( pzomcstn \\). Then \\( jsmbwzoh^{2}+gpldxeio^{2}=vutnqrsa^{2} \\) and \\( rypqnesh=gpldxeio / jsmbwzoh \\).\n\nThe intersection set \\( vhgalezr \\) of \\( \\Pi \\) and the torus certainly contains the following six points: four points of the form ( \\( 0, \\pm vutnqrsa \\pm gpldxeio, 0 \\) ) on the \\( hjgrksla \\) axis, and the two points \\( pzomcstn \\) and \\( pzomcstn^{\\prime} \\). We note that the set \\( vhgalezr \\) is also symmetric in the \\( hjgrksla \\)-axis. Hence if \\( vhgalezr \\) consists of two circles, they must be the circles of radius \\( vutnqrsa \\) and centers ( \\( 0, \\pm gpldxeio, 0 \\) ) lying in the plane \\( \\Pi \\). (See figure.)\n\nParametric equations for the first of these circles are\n\\[\n\\begin{array}{l}\nhjgrksla=gpldxeio+vutnqrsa \\sin frewpdsk \\\\\nqzxwvtnp=jsmbwzoh \\cos frewpdsk \\\\\nkovmlpqa=gpldxeio \\cos frewpdsk\n\\end{array}\n\\]\n\nTo check this, note that these values satisfy both \\( qzxwvtnp^{2}+(hjgrksla-gpldxeio)^{2}+kovmlpqa^{2}=vutnqrsa^{2} \\) and \\( kovmlpqa=rypqnesh\\,qzxwvtnp \\), so (1) describes a closed curve that lies on the sphere of radius \\( vutnqrsa \\) about \\( (0, gpldxeio, 0) \\) and on the plane \\( kovmlpqa=rypqnesh\\,qzxwvtnp \\). (To get the second circle just reverse the sign of \\( gpldxeio \\).)\n\nNext we show that these circles lie on the torus. For any point on (1) we have\n\\[\n\\begin{aligned}\nqzxwvtnp^{2}+hjgrksla^{2} & =jsmbwzoh^{2} \\cos ^{2} frewpdsk+gpldxeio^{2}+2\\,vutnqrsa\\,gpldxeio \\sin frewpdsk+vutnqrsa^{2} \\sin ^{2} frewpdsk \\\\\n& =\\left(vutnqrsa^{2}-gpldxeio^{2}\\right) \\cos ^{2} frewpdsk+gpldxeio^{2}+2\\,vutnqrsa\\,gpldxeio \\sin frewpdsk+vutnqrsa^{2} \\sin ^{2} frewpdsk \\\\\n& =vutnqrsa^{2}+2\\,vutnqrsa\\,gpldxeio \\sin frewpdsk+gpldxeio^{2} \\sin ^{2} frewpdsk=(vutnqrsa+gpldxeio \\sin frewpdsk)^{2}\n\\end{aligned}\n\\]\n\nSince \\( vutnqrsa>gpldxeio,\\,vutnqrsa+gpldxeio \\sin frewpdsk>0 \\); so \\( \\sqrt{qzxwvtnp^{2}}+hjgrksla^{2}=vutnqrsa+gpldxeio \\sin frewpdsk \\). Hence\n\\[\n\\left[\\sqrt{qzxwvtnp^{2}}+hjgrksla^{2}-vutnqrsa\\right]^{2}+kovmlpqa^{2}=gpldxeio^{2} \\sin ^{2} frewpdsk+gpldxeio^{2} \\cos ^{2} frewpdsk=gpldxeio^{2}\n\\]\nwhich is the defining equation of the torus.\nThus the first circle lies on the torus. That the second circle does also follows by similar algebra, or by noting that the two circles are interchanged by a reflection in the \\( qzxwvtnp\\,kovmlpqa \\)-plane.\n\nRemark. This solution is not quite complete, because it only shows that \\( vhgalezr \\) contains the two circles; conceivably there are further points in the intersection set \\( vhgalezr \\), although it is intuitively evident that there are not.\n\nA complete solution can be obtained by purely algebraic manipulations. If the radical is eliminated from the original equation for the torus, we obtain the equivalent equation\n\\[\n\\left(qzxwvtnp^{2}+hjgrksla^{2}+kovmlpqa^{2}+vutnqrsa^{2}-gpldxeio^{2}\\right)^{2}=4\\,vutnqrsa^{2}\\,qzxwvtnp^{2}+4\\,vutnqrsa^{2}\\,hjgrksla^{2}\n\\]\n\nThis can be rewritten as\n\\[\n\\left(qzxwvtnp^{2}+hjgrksla^{2}+kovmlpqa^{2}+gpldxeio^{2}-vutnqrsa^{2}\\right)^{2}-4\\,gpldxeio^{2}\\,hjgrksla^{2}=4\\,gpldxeio^{2}\\,qzxwvtnp^{2}-4\\,jsmbwzoh^{2}\\,kovmlpqa^{2}\n\\]\nand then as\n\\[\n\\begin{aligned}\n\\left(qzxwvtnp^{2}+(hjgrksla-gpldxeio)^{2}+kovmlpqa^{2}-vutnqrsa^{2}\\right)\\left(qzxwvtnp^{2}+(hjgrksla+gpldxeio)^{2}+\\right. & \\left.kovmlpqa^{2}-vutnqrsa^{2}\\right) \\\\\n& =4(gpldxeio\\,qzxwvtnp-jsmbwzoh\\,kovmlpqa)(gpldxeio\\,qzxwvtnp+jsmbwzoh\\,kovmlpqa)\n\\end{aligned}\n\\]\n\nFrom this equation it appears that any point on both the torus and the plane \\( gpldxeio\\,qzxwvtnp-jsmbwzoh\\,kovmlpqa=0 \\) is also on one of the two spheres\n\\[\nqzxwvtnp^{2}+(hjgrksla-gpldxeio)^{2}+kovmlpqa^{2}=vutnqrsa^{2} \\text { and } qzxwvtnp^{2}+(hjgrksla+gpldxeio)^{2}+kovmlpqa^{2}=vutnqrsa^{2}\n\\]\n\nConversely, any point on the plane and either sphere lies also on the torus. Thus the intersection set \\( vhgalezr \\) is exactly the union of the two circles.\n\nRemark. This problem is discussed in H. S. M. Coxeter, Introduction to Geometry, Wiley, New York, 1961, pages 132-133. References to other papers may be found in Coxeter's book."
    },
    "kernel_variant": {
      "question": "Let a>b>0 be real numbers.  Put the origin O at the centre of the torus of revolution obtained by rotating the circle of radius b with centre (a,0,0) that lies in the x–y–plane about the y–axis.  The torus is therefore the set\n\n            T:=\\{(x,y,z)\\in\\mathbb R^{3}\\mid (\\sqrt{x^{2}+z^{2}}-a)^{2}+y^{2}=b^{2}\\}.   \\tag{1}\n\nProve that every plane that\n   (i) passes through the origin and\n   (ii) is tangent to the torus T\nmeets the torus in exactly two (non-disjoint) circles.  Both circles lie in the plane of tangency and intersect each other in the two points where the plane touches the torus.\n\n(The two circles are the classical Villarceau circles.)",
      "solution": "Throughout we put\n            A:=a\\;(>0),\\;\\;B:=b\\;(0<B<A),\\;\\;C:=\\sqrt{A^{2}-B^{2}}.   \\tag{2}\n\n1.  A convenient description of a tangent plane.\n\nA point Q=(x_{0},y_{0},z_{0}) lies on the torus T if and only if\n           (\\sqrt{x_{0}^{2}+z_{0}^{2}}-A)^{2}+y_{0}^{2}=B^{2}.           \\tag{3}\nThe (non-zero) normal vector of the torus at Q is the gradient of the quartic polynomial\n           F(x,y,z):=(x^{2}+y^{2}+z^{2}+A^{2}-B^{2})^{2}-4A^{2}(x^{2}+z^{2}) \\tag{4}\nwhose zero–set is T.  Explicitly\n           \\nabla F(Q)=2\\,(S+A^{2}-B^{2})\\,(x_{0},y_{0},z_{0})-4A^{2}(x_{0},0,z_{0}),\\qquad S:=x_{0}^{2}+y_{0}^{2}+z_{0}^{2}. \\tag{5}\n\nBecause the plane we are looking for is required to pass through the origin, its equation must have the form\n           g_{x}x+g_{y}y+g_{z}z=0,                         \\tag{6}\nwhere g=(g_{x},g_{y},g_{z}) is some non–zero vector parallel to \\nabla F(Q).\nThe point Q itself must satisfy (6), hence g\\cdot Q=0.  Writing out g (use (5)) and performing an elementary elimination of S one obtains the single linear relation\n           Bx_{0}-Cz_{0}=0.                                \\tag{7}\nConsequently every tangent plane through O has an equation of the form\n           \\Pi:\\;Bx-Cz=0.                                  \\tag{8}\n(The sign of the right-hand side is irrelevant.)  Up to the rotational symmetry of the torus around the y–axis this is the *only* possibility.\n\nBecause B,\\,C\\neq 0 the line of intersection of \\Pi with the xz–plane is uniquely fixed; hence (8) determines \\Pi completely.\n\n2.  Two quadratic auxiliary polynomials.\n\nIntroduce\n           H_{\\pm}(x,y,z):=x^{2}+(y\\mp B)^{2}+z^{2}-A^{2}.             \\tag{9}\nThe zero–set of H_{+} is the sphere S_{+} with centre (0, B,0) and radius A; analogously H_{-}^{-1}(0)=S_{-} is the sphere with centre (0,-B,0) and the same radius A.\n\n3.  The torus restricted to the plane \\Pi.\n\nOn the plane \\Pi we have Bx=Cz, hence x=\\frac{C}{B}z and\n           x^{2}+z^{2}=\\bigl(\\tfrac{C^{2}}{B^{2}}+1\\bigr)z^{2}=\\frac{A^{2}}{B^{2}}\\,z^{2}. \\tag{10}\nSubstituting (10) into the defining equation (1) of the torus and rearranging one obtains after a short calculation\n           \\bigl[x^{2}+(y-B)^{2}+z^{2}-A^{2}\\bigr]\\,\\bigl[x^{2}+(y+B)^{2}+z^{2}-A^{2}\\bigr]=0. \\tag{11}\nBut the two bracketed expressions are exactly H_{+} and H_{-}.  Therefore\n           T\\cap\\Pi=\\bigl(S_{+}\\cap\\Pi\\bigr)\\;\\cup\\;\\bigl(S_{-}\\cap\\Pi\\bigr). \\tag{12}\n\n4.  Geometry of the two components.\n\nBecause the centres (0,\\pm B,0) of the two spheres lie on \\Pi (they satisfy (8)), the intersections\n           \\Gamma_{+}:=\\Pi\\cap S_{+},\\qquad \\Gamma_{-}:=\\Pi\\cap S_{-} \\tag{13}\nare circles of radius A that lie completely inside the plane \\Pi.  The distance of the two centres equals 2B<2A, hence the two circles meet in exactly two points.  A direct check shows that those two points are the unique solutions of (7) *and* y_{0}=0, i.e. precisely the two points where \\Pi is tangent to the torus.\n\n5.  No other points in the intersection.\n\nConversely, if a point P belongs to \\Pi and to S_{+} (or S_{-}) then it satisfies (11) and therefore the original torus equation (1); hence P lies on T.  Together with (12) this proves\n           T\\cap\\Pi=\\Gamma_{+}\\cup\\Gamma_{-}.                  \\tag{14}\n\n6.  Conclusion.\n\nEvery plane through the origin that is tangent to the standard torus meets the torus exactly in the two circles (13).  Both circles lie in the tangent plane, have radius A, are centred at (0,\\pm B,0), and intersect each other in the two points of tangency.  These two circles are the Villarceau circles. ▢",
      "_meta": {
        "core_steps": [
          "Exploit rotational symmetry to fix coordinates and write the torus as (√(x²+y²)−a)²+z²=b² with a>b>0",
          "Describe every plane through the center that is tangent to the torus as b x−c z=0 where c²=a²−b²",
          "Insert that plane equation into the torus equation, clear the square-root and obtain a factorization into two quadratic factors",
          "Identify each quadratic factor with a sphere centred at (0,±b,0); the plane meets each sphere in a circle",
          "Conclude the intersection set equals exactly those two circles (no extra points because the factorization is exact)"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Choice of coordinate axis taken as the axis of rotational symmetry",
            "original": "z–axis"
          },
          "slot2": {
            "description": "Names and ordering of the torus parameters subject to the inequality larger>smaller>0",
            "original": "a>b>0"
          },
          "slot3": {
            "description": "Letter used for distance from center to tangent point satisfying c² = a²−b²",
            "original": "c"
          },
          "slot4": {
            "description": "Sign/orientation of the tangent plane equation; either b x−c z=0 or b x+c z=0 (equivalently z=±λx)",
            "original": "b x−c z=0"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}