Let us consider the set <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega (\triangle ABC)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>▵</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of all tetrahedra ABCD with a given non-degenerate base ABC in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {E}}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> and D lying outside the plane ABC. Let us denote by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma (\triangle ABC)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">(</mo> <mi>▵</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left\{ \Bigl (\cos {\overline{\alpha }},\cos {\overline{\beta }},\cos {\overline{\gamma }} \Bigr )\in {\mathbb {R}}^3\,|\, ABCD \in \Omega (\triangle ABC)\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mo>cos</mo> <mover> <mi>α</mi> <mo>¯</mo> </mover> <mo>,</mo> <mo>cos</mo> <mover> <mi>β</mi> <mo>¯</mo> </mover> <mo>,</mo> <mo>cos</mo> <mover> <mi>γ</mi> <mo>¯</mo> </mover> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mrow> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>▵</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfenced> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\overline{\alpha }}=\angle BDC$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>α</mi> <mo>¯</mo> </mover> <mo>=</mo> <mo>∠</mo> <mi>B</mi> <mi>D</mi> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\overline{\beta }}=\angle ADC$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>β</mi> <mo>¯</mo> </mover> <mo>=</mo> <mo>∠</mo> <mi>A</mi> <mi>D</mi> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\overline{\gamma }}=\angle ADB$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>γ</mi> <mo>¯</mo> </mover> <mo>=</mo> <mo>∠</mo> <mi>A</mi> <mi>D</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>. The paper is devoted to the problem of determining the closure of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Sigma (\triangle ABC)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">(</mo> <mi>▵</mi> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\mathbb {R}}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> and its boundary.

On Face Angles of Tetrahedra with a Given Base

期刊论文
发表日期：

2026年3月6日

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E. V. Nikitenko,
Yu.G. Nikonorov

摘要

Let us consider the set $\Omega (\triangle ABC)$ $Ω (▵ A B C)$ of all tetrahedra ABCD with a given non-degenerate base ABC in ${\mathbb {E}}^3$ $E^{3}$ and D lying outside the plane ABC. Let us denote by $\Sigma (\triangle ABC)$ $Σ (▵ A B C)$ the set $\left\{ \Bigl (\cos {\overline{\alpha }},\cos {\overline{\beta }},\cos {\overline{\gamma }} \Bigr )\in {\mathbb {R}}^3\,|\, ABCD \in \Omega (\triangle ABC)\right\} $ $\{(cos \bar{α}, cos \bar{β}, cos \bar{γ}) \in R^{3} | A B C D \in Ω (▵ A B C)\}$ , where ${\overline{\alpha }}=\angle BDC$ $\bar{α} = ∠ B D C$ , ${\overline{\beta }}=\angle ADC$ $\bar{β} = ∠ A D C$ , and ${\overline{\gamma }}=\angle ADB$ $\bar{γ} = ∠ A D B$ . The paper is devoted to the problem of determining the closure of $\Sigma (\triangle ABC)$ $Σ (▵ A B C)$ in ${\mathbb {R}}^3$ $R^{3}$ and its boundary.