Approximation with a hybrid DD Stancu operator: Applications to coronary vessel visualization, image representation, and geometric modeling

Emine Güven; Nazmiye Gönül Bilgin; Emine Güven; Nazmiye Gönül Bilgin

doi:10.3934/math.2026902

AIMS Mathematics

2026, Volume 11, Issue 7: 22277-22315. doi: 10.3934/math.2026902

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Research article

Approximation with a hybrid DD Stancu operator: Applications to coronary vessel visualization, image representation, and geometric modeling

Emine Güven ¹,
Nazmiye Gönül Bilgin ^{2
,
,}

1.
Graduate School of Natural and Applied Sciences, Zonguldak Bülent Ecevit University, Turkey
2.
Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, Turkey

Received: 11 May 2026 Revised: 06 July 2026 Accepted: 15 July 2026 Published: 24 July 2026
MSC : 41A35, 41A36, 65D17, 68U07

This study presents a generalization of the DD Stancu operator reported in the literature by combining it with the lambda Bernstein operator. Using the basis functions of the proposed operator, geometric modeling applications related to vascular visualization, botanical form representation, and entomological form modeling are presented; thus, the role of approximation theory in mathematical modeling and the aforementioned fields is emphasized. The biological modeling and botanical illustration generated using the basis functions of the operator demonstrate the effectiveness of the method in representing organic shapes by enabling the λ-parameter to capture with high precision the natural forms of leaves and butterflies, their internal curves, and venation structures. The model, developed to represent structural changes such as narrowing and widening in vascular geometry, was designed solely as a geometric visualization tool rather than as a validated medical simulation or diagnostic method, and is made available for further studies.
- Bézier-type curve,
- λ-modification,
- double summation operator,
- linear positive operators,
- approximation theory,
- geometric modeling
Citation: Emine Güven, Nazmiye Gönül Bilgin. Approximation with a hybrid DD Stancu operator: Applications to coronary vessel visualization, image representation, and geometric modeling[J]. AIMS Mathematics, 2026, 11(7): 22277-22315. doi: 10.3934/math.2026902

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Abstract

This study presents a generalization of the DD Stancu operator reported in the literature by combining it with the lambda Bernstein operator. Using the basis functions of the proposed operator, geometric modeling applications related to vascular visualization, botanical form representation, and entomological form modeling are presented; thus, the role of approximation theory in mathematical modeling and the aforementioned fields is emphasized. The biological modeling and botanical illustration generated using the basis functions of the operator demonstrate the effectiveness of the method in representing organic shapes by enabling the λ-parameter to capture with high precision the natural forms of leaves and butterflies, their internal curves, and venation structures. The model, developed to represent structural changes such as narrowing and widening in vascular geometry, was designed solely as a geometric visualization tool rather than as a validated medical simulation or diagnostic method, and is made available for further studies.

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