<p>In this paper, we define <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids of spherical curves in spherical space. The relationship between <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-evolutoids of spherical curves is established. We generalize the notion to the category of spherical frontals and find that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids of spherical frontals may have singularities, while <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids of spherical unit speed curves are regular. Then, we investigate properties of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids of spherical frontals by using the singularity theory. From the viewpoint of the contact geometry, we prove the singularities of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids are deeply related to the order of contact between <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-orthotomicoids and specific circles on the unit sphere. Finally, we provide some examples to demonstrate main results.</p>

$\theta $-Orthotomicoids of curves on the unit sphere

期刊论文
发表日期：

2026年8月4日

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Yongqiao Wang,
Lili Wang,
Yuan Chang

摘要

In this paper, we define $\theta $ $θ$ -orthotomicoids of spherical curves in spherical space. The relationship between $\theta $ $θ$ -orthotomicoids and $\theta $ $θ$ -evolutoids of spherical curves is established. We generalize the notion to the category of spherical frontals and find that $\theta $ $θ$ -orthotomicoids of spherical frontals may have singularities, while $\theta $ $θ$ -orthotomicoids of spherical unit speed curves are regular. Then, we investigate properties of $\theta $ $θ$ -orthotomicoids of spherical frontals by using the singularity theory. From the viewpoint of the contact geometry, we prove the singularities of $\theta $ $θ$ -orthotomicoids are deeply related to the order of contact between $\theta $ $θ$ -orthotomicoids and specific circles on the unit sphere. Finally, we provide some examples to demonstrate main results.

\(\theta \)-Orthotomicoids of curves on the unit sphere

摘要