{
  "index": "1955-B-3",
  "type": "GEO",
  "tag": [
    "GEO",
    "ANA"
  ],
  "difficulty": "",
  "question": "3. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)",
  "solution": "Solution. We shall denote the spherical distance between two points \\( A \\) and \\( B \\) by \\( \\rho(A, B) \\) and the ordinary Euclidean distance by \\( |A B| \\).\n\nSuppose \\( X \\mapsto X^{\\prime} \\) is a distance-preserving map of a spherical cap into the plane. Let \\( A, B, C, D \\) be four points of the cap such that \\( A B C D \\) is a Euclidean square in three-space. Note that a spherical cap must contain such a set. Then \\( \\rho(A, B)=\\rho(B, C)=\\rho(C, D)=\\rho(D, A) \\) and \\( \\rho(A, C) \\) \\( =\\rho(B, D) \\). These relations imply that\n\\[\n\\left|A^{\\prime} B^{\\prime}\\right|=\\left|B^{\\prime} C^{\\prime}\\right|=\\left|C^{\\prime} D^{\\prime}\\right|=\\left|D^{\\prime} A^{\\prime}\\right| \\text { and }\\left|A^{\\prime} C^{\\prime}\\right|=\\left|B^{\\prime} D^{\\prime}\\right| .\n\\]\n\nHence \\( A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime} \\) is a square, and \\( \\left|A^{\\prime} C^{\\prime}\\right|=\\sqrt{2}\\left|A^{\\prime} B^{\\prime}\\right| \\). Since distances are preserved by the mapping, \\( \\rho(A, C)=\\sqrt{2} \\rho(A, B) \\). Now \\( |A C|= \\) \\( \\sqrt{ } 2|A B| \\) because \\( A B C D \\) is a square, so\n\\[\n\\frac{\\rho(A, C)}{|A C|}=\\frac{\\rho(A, B)}{|A B|}\n\\]\n\nHowever, \\( \\rho(X . Y) /|X Y|=\\theta / \\sin \\theta \\), where \\( \\theta \\) is half the central angle of the arc \\( X Y \\). Now \\( \\theta / \\sin \\theta \\) is a strictly increasing function of \\( \\theta \\) and hence of \\( |X Y| \\) and \\( |A C|>|A B| \\); therefore, (1) is impossible. This contradiction proves that no distance-preserving map of a spherical cap into a plane exists.",
  "vars": [
    "A",
    "B",
    "C",
    "D",
    "X",
    "Y"
  ],
  "params": [
    "\\\\rho",
    "\\\\theta"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "pointa",
        "B": "pointb",
        "C": "pointc",
        "D": "pointd",
        "X": "pointx",
        "Y": "pointy",
        "\\rho": "spheredist",
        "\\theta": "centralang"
      },
      "question": "3. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)",
      "solution": "Solution. We shall denote the spherical distance between two points \\( pointa \\) and \\( pointb \\) by \\( spheredist(pointa, pointb) \\) and the ordinary Euclidean distance by \\(|pointa pointb|\\).\n\nSuppose \\( pointx \\mapsto pointx^{\\prime} \\) is a distance-preserving map of a spherical cap into the plane. Let \\( pointa, pointb, pointc, pointd \\) be four points of the cap such that \\( pointa pointb pointc pointd \\) is a Euclidean square in three-space. Note that a spherical cap must contain such a set. Then \\( spheredist(pointa, pointb)=spheredist(pointb, pointc)=spheredist(pointc, pointd)=spheredist(pointd, pointa) \\) and \\( spheredist(pointa, pointc)=spheredist(pointb, pointd) \\). These relations imply that\n\\[\n\\left|pointa^{\\prime} pointb^{\\prime}\\right|=\\left|pointb^{\\prime} pointc^{\\prime}\\right|=\\left|pointc^{\\prime} pointd^{\\prime}\\right|=\\left|pointd^{\\prime} pointa^{\\prime}\\right| \\text { and }\\left|pointa^{\\prime} pointc^{\\prime}\\right|=\\left|pointb^{\\prime} pointd^{\\prime}\\right| .\n\\]\n\nHence \\( pointa^{\\prime} pointb^{\\prime} pointc^{\\prime} pointd^{\\prime} \\) is a square, and \\( \\left|pointa^{\\prime} pointc^{\\prime}\\right|=\\sqrt{2}\\left|pointa^{\\prime} pointb^{\\prime}\\right| \\). Since distances are preserved by the mapping, \\( spheredist(pointa, pointc)=\\sqrt{2}\\,spheredist(pointa, pointb) \\). Now \\( |pointa pointc|= \\sqrt{ } 2|pointa pointb| \\) because \\( pointa pointb pointc pointd \\) is a square, so\n\\[\n\\frac{spheredist(pointa, pointc)}{|pointa pointc|}=\\frac{spheredist(pointa, pointb)}{|pointa pointb|}\n\\]\n\nHowever, \\( spheredist(pointx . pointy) /|pointx pointy|=centralang / \\sin centralang \\), where \\( centralang \\) is half the central angle of the arc \\( pointx pointy \\). Now \\( centralang / \\sin centralang \\) is a strictly increasing function of \\( centralang \\) and hence of \\( |pointx pointy| \\) and \\( |pointa pointc|>|pointa pointb| \\); therefore, (1) is impossible. This contradiction proves that no distance-preserving map of a spherical cap into a plane exists."
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "sandcastle",
        "B": "driftwood",
        "C": "lighthouse",
        "D": "boardwalk",
        "X": "seashells",
        "Y": "shipwreck",
        "\\rho": "boardlength",
        "\\theta": "stormfront"
      },
      "question": "3. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)",
      "solution": "Solution. We shall denote the spherical distance between two points \\( sandcastle \\) and \\( driftwood \\) by \\( boardlength(sandcastle, driftwood) \\) and the ordinary Euclidean distance by \\( |sandcastle driftwood| \\).\n\nSuppose \\( seashells \\mapsto seashells^{\\prime} \\) is a distance-preserving map of a spherical cap into the plane. Let \\( sandcastle, driftwood, lighthouse, boardwalk \\) be four points of the cap such that \\( sandcastle driftwood lighthouse boardwalk \\) is a Euclidean square in three-space. Note that a spherical cap must contain such a set. Then \\( boardlength(sandcastle, driftwood)=boardlength(driftwood, lighthouse)=boardlength(lighthouse, boardwalk)=boardlength(boardwalk, sandcastle) \\) and \\( boardlength(sandcastle, lighthouse) \\) \\( =boardlength(driftwood, boardwalk) \\). These relations imply that\n\\[\n\\left|sandcastle^{\\prime} driftwood^{\\prime}\\right|=\\left|driftwood^{\\prime} lighthouse^{\\prime}\\right|=\\left|lighthouse^{\\prime} boardwalk^{\\prime}\\right|=\\left|boardwalk^{\\prime} sandcastle^{\\prime}\\right| \\text { and }\\left|sandcastle^{\\prime} lighthouse^{\\prime}\\right|=\\left|driftwood^{\\prime} boardwalk^{\\prime}\\right| .\n\\]\n\nHence \\( sandcastle^{\\prime} driftwood^{\\prime} lighthouse^{\\prime} boardwalk^{\\prime} \\) is a square, and \\( \\left|sandcastle^{\\prime} lighthouse^{\\prime}\\right|=\\sqrt{2}\\left|sandcastle^{\\prime} driftwood^{\\prime}\\right| \\). Since distances are preserved by the mapping, \\( boardlength(sandcastle, lighthouse)=\\sqrt{2} boardlength(sandcastle, driftwood) \\). Now \\( |sandcastle lighthouse|= \\) \\( \\sqrt{ } 2|sandcastle driftwood| \\) because \\( sandcastle driftwood lighthouse boardwalk \\) is a square, so\n\\[\n\\frac{boardlength(sandcastle, lighthouse)}{|sandcastle lighthouse|}=\\frac{boardlength(sandcastle, driftwood)}{|sandcastle driftwood|}\n\\]\n\nHowever, \\( boardlength(seashells . shipwreck) /|seashells shipwreck|=stormfront / \\sin stormfront \\), where \\( stormfront \\) is half the central angle of the arc \\( seashells shipwreck \\). Now \\( stormfront / \\sin stormfront \\) is a strictly increasing function of \\( stormfront \\) and hence of \\( |seashells shipwreck| \\) and \\( |sandcastle lighthouse|>|sandcastle driftwood| \\); therefore, (1) is impossible. This contradiction proves that no distance-preserving map of a spherical cap into a plane exists."
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "nonpoint",
        "B": "blankspace",
        "C": "emptyset",
        "D": "voidregion",
        "X": "nowherepos",
        "Y": "nothingloc",
        "\\rho": "contactness",
        "\\theta": "straightness"
      },
      "question": "3. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)",
      "solution": "Solution. We shall denote the spherical distance between two points \\( nonpoint \\) and \\( blankspace \\) by \\( contactness(nonpoint, blankspace) \\) and the ordinary Euclidean distance by \\( |nonpoint blankspace| \\).\n\nSuppose \\( nowherepos \\mapsto nowherepos^{\\prime} \\) is a distance-preserving map of a spherical cap into the plane. Let \\( nonpoint, blankspace, emptyset, voidregion \\) be four points of the cap such that \\( nonpoint blankspace emptyset voidregion \\) is a Euclidean square in three-space. Note that a spherical cap must contain such a set. Then \\( contactness(nonpoint, blankspace)=contactness(blankspace, emptyset)=contactness(emptyset, voidregion)=contactness(voidregion, nonpoint) \\) and \\( contactness(nonpoint, emptyset) \\) \\( =contactness(blankspace, voidregion) \\). These relations imply that\n\\[\n\\left|nonpoint^{\\prime} blankspace^{\\prime}\\right|=\\left|blankspace^{\\prime} emptyset^{\\prime}\\right|=\\left|emptyset^{\\prime} voidregion^{\\prime}\\right|=\\left|voidregion^{\\prime} nonpoint^{\\prime}\\right| \\text { and }\\left|nonpoint^{\\prime} emptyset^{\\prime}\\right|=\\left|blankspace^{\\prime} voidregion^{\\prime}\\right| .\n\\]\n\nHence \\( nonpoint^{\\prime} blankspace^{\\prime} emptyset^{\\prime} voidregion^{\\prime} \\) is a square, and \\( \\left|nonpoint^{\\prime} emptyset^{\\prime}\\right|=\\sqrt{2}\\left|nonpoint^{\\prime} blankspace^{\\prime}\\right| \\). Since distances are preserved by the mapping, \\( contactness(nonpoint, emptyset)=\\sqrt{2} contactness(nonpoint, blankspace) \\). Now \\( |nonpoint emptyset|= \\) \\( \\sqrt{ } 2|nonpoint blankspace| \\) because \\( nonpoint blankspace emptyset voidregion \\) is a square, so\n\\[\n\\frac{contactness(nonpoint, emptyset)}{|nonpoint emptyset|}=\\frac{contactness(nonpoint, blankspace)}{|nonpoint blankspace|}\n\\]\n\nHowever, \\( contactness(nowherepos . nothingloc) /|nowherepos nothingloc|=straightness / \\sin straightness \\), where \\( straightness \\) is half the central angle of the arc \\( nowherepos nothingloc \\). Now \\( straightness / \\sin straightness \\) is a strictly increasing function of \\( straightness \\) and hence of \\( |nowherepos nothingloc| \\) and \\( |nonpoint emptyset|>|nonpoint blankspace| \\); therefore, (1) is impossible. This contradiction proves that no distance-preserving map of a spherical cap into a plane exists."
    },
    "garbled_string": {
      "map": {
        "A": "qzxwvtnp",
        "B": "hjgrksla",
        "C": "mpfldvqe",
        "D": "snbxrywc",
        "X": "vltpcrhe",
        "Y": "kdmqsgow",
        "\\\\rho": "\\\\qprnfghj",
        "\\\\theta": "\\\\mnvbcxzi"
      },
      "question": "3. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)",
      "solution": "Solution. We shall denote the spherical distance between two points \\( qzxwvtnp \\) and \\( hjgrksla \\) by \\( \\qprnfghj(qzxwvtnp, hjgrksla) \\) and the ordinary Euclidean distance by \\( |qzxwvtnp hjgrksla| \\).\n\nSuppose \\( vltpcrhe \\mapsto vltpcrhe^{\\prime} \\) is a distance-preserving map of a spherical cap into the plane. Let \\( qzxwvtnp, hjgrksla, mpfldvqe, snbxrywc \\) be four points of the cap such that \\( qzxwvtnp hjgrksla mpfldvqe snbxrywc \\) is a Euclidean square in three-space. Note that a spherical cap must contain such a set. Then \\( \\qprnfghj(qzxwvtnp, hjgrksla)=\\qprnfghj(hjgrksla, mpfldvqe)=\\qprnfghj(mpfldvqe, snbxrywc)=\\qprnfghj(snbxrywc, qzxwvtnp) \\) and \\( \\qprnfghj(qzxwvtnp, mpfldvqe)=\\qprnfghj(hjgrksla, snbxrywc) \\). These relations imply that\n\\[\n\\left|qzxwvtnp^{\\prime} hjgrksla^{\\prime}\\right|=\\left|hjgrksla^{\\prime} mpfldvqe^{\\prime}\\right|=\\left|mpfldvqe^{\\prime} snbxrywc^{\\prime}\\right|=\\left|snbxrywc^{\\prime} qzxwvtnp^{\\prime}\\right| \\text { and }\\left|qzxwvtnp^{\\prime} mpfldvqe^{\\prime}\\right|=\\left|hjgrksla^{\\prime} snbxrywc^{\\prime}\\right| .\n\\]\n\nHence \\( qzxwvtnp^{\\prime} hjgrksla^{\\prime} mpfldvqe^{\\prime} snbxrywc^{\\prime} \\) is a square, and \\( \\left|qzxwvtnp^{\\prime} mpfldvqe^{\\prime}\\right|=\\sqrt{2}\\left|qzxwvtnp^{\\prime} hjgrksla^{\\prime}\\right| \\). Since distances are preserved by the mapping, \\( \\qprnfghj(qzxwvtnp, mpfldvqe)=\\sqrt{2} \\qprnfghj(qzxwvtnp, hjgrksla) \\). Now \\( |qzxwvtnp mpfldvqe|= \\sqrt{ } 2|qzxwvtnp hjgrksla| \\) because \\( qzxwvtnp hjgrksla mpfldvqe snbxrywc \\) is a square, so\n\\[\n\\frac{\\qprnfghj(qzxwvtnp, mpfldvqe)}{|qzxwvtnp mpfldvqe|}=\\frac{\\qprnfghj(qzxwvtnp, hjgrksla)}{|qzxwvtnp hjgrksla|}\n\\]\n\nHowever, \\( \\qprnfghj(vltpcrhe . kdmqsgow) /|vltpcrhe kdmqsgow|=\\mnvbcxzi / \\sin \\mnvbcxzi \\), where \\( \\mnvbcxzi \\) is half the central angle of the arc \\( vltpcrhe kdmqsgow \\). Now \\( \\mnvbcxzi / \\sin \\mnvbcxzi \\) is a strictly increasing function of \\( \\mnvbcxzi \\) and hence of \\( |vltpcrhe kdmqsgow| \\) and \\( |qzxwvtnp mpfldvqe|>|qzxwvtnp hjgrksla| \\); therefore, (1) is impossible. This contradiction proves that no distance-preserving map of a spherical cap into a plane exists."
    },
    "kernel_variant": {
      "question": "Let \\(S\\subset \\,\\mathbb R^{3}\\) be part of the unit sphere obtained by intersecting the sphere with some open half-space whose bounding plane meets the sphere (a \"spherical cap\").  A map \\(f:S\\to\\mathbb R^{2}\\) is called distance–preserving if, for every pair of points \\(X,Y\\in S\\), the great–circle (intrinsic) distance \\(\\rho(X,Y)\\) equals the ordinary Euclidean distance \\(|f(X)f(Y)|\\) in the plane.\n\nProve that no such distance–preserving map exists.\n\n(Hint: inside every spherical cap one can find four points that form a  \\(1\\!:\\!2\\) Euclidean rectangle in three–space.)",
      "solution": "Suppose, to the contrary, that there is a distance-preserving map f from the spherical cap S into the plane.  We denote by ρ(X,Y) the great-circle distance on the unit sphere and by |XY| the ordinary chord (Euclidean) distance in ℝ³ or, for images, in ℝ².\n\n1. Inside the cap pick four points A, B, C, D that form a small Euclidean square in ℝ³ of side ℓ (ℓ chosen small enough that the entire square lies in the cap).  Then in space\n   |AB| = |BC| = |CD| = |DA| = ℓ,\n   |AC| = |BD| = √2 ℓ.\n\n2. Since f is distance-preserving (for great-circle distance),\n   |f(A)f(B)| = ρ(A,B),\n   |f(B)f(C)| = ρ(B,C),\n   |f(C)f(D)| = ρ(C,D),\n   |f(D)f(A)| = ρ(D,A),\nand\n   |f(A)f(C)| = ρ(A,C),\n   |f(B)f(D)| = ρ(B,D).\nBut in the original square ρ(A,B)=ρ(B,C)=ρ(C,D)=ρ(D,A) and ρ(A,C)=ρ(B,D).  Hence in the plane the quadrilateral f(A)f(B)f(C)f(D) has four equal sides and equal diagonals, so it is itself a square.  In particular,\n   |f(A)f(C)| = √2 |f(A)f(B)|.\nTranslating back to spherical distances gives\n   ρ(A,C) = √2 ρ(A,B).\n\n3. Relate each spherical distance to the corresponding chord length.  If the half–central angle for X,Y is θ so that the arc length is 2θ, then\n   ρ(X,Y) = 2θ,    |XY| = 2 sin θ,\n   ⇒ ρ(X,Y)/|XY| = θ/ sin θ =: g(θ).\nIt is well known (and easily checked by differentiation) that g(θ) is strictly increasing for 0<θ<π.\n\n4. In our square in ℝ³, |AC|=√2 ℓ > ℓ = |AB|, so the corresponding half–angles satisfy θ_{AC} > θ_{AB}.  Hence\n   g(θ_{AC}) > g(θ_{AB}).\nBut by definition,\n   g(θ_{AC}) = ρ(A,C)/|AC|,    g(θ_{AB}) = ρ(A,B)/|AB|.\nUsing ρ(A,C)=√2 ρ(A,B) and |AC|=√2 |AB| we would have\n   ρ(A,C)/|AC| = (√2 ρ(A,B))/(√2 |AB|) = ρ(A,B)/|AB|,\ncontradicting the strict inequality g(θ_{AC}) > g(θ_{AB}).\n\nTherefore no distance-preserving map of the spherical cap into the plane can exist.  ■",
      "_meta": {
        "core_steps": [
          "Assume an isometry from a spherical cap to the plane and seek contradiction.",
          "Pick two chords coming from a fixed planar figure whose Euclidean lengths have a known constant ratio (>1).",
          "Because the map preserves distance, the corresponding spherical (arc) lengths must share that same constant ratio.",
          "Relate arc length to chord length via ρ/|XY| = θ/ sin θ and note this quotient increases strictly with |XY|.",
          "Since the longer chord should give a larger quotient, equality of the two quotients is impossible, contradicting the assumed isometry."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Specific planar figure chosen only to guarantee two distinct chord lengths with a fixed ratio (e.g., square, rectangle, right-triangle hypotenuse vs. leg).",
            "original": "Euclidean square A B C D"
          },
          "slot2": {
            "description": "Numeric value of the ratio between the two chord lengths, determined by the chosen figure; any constant >1 would work.",
            "original": "√2"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}