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Approximation of Planar Curves for Computer Aided Designs

Thesis Info

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Author

Shakeel, Ayesha

Program

PhD

Institute

University of the Punjab

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12423/1/Ayesha%20Shakeel_Maths_2019_UoPunjab_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725528671

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Computer Aided Design (CAD) is used for the manufacture and analysis of designs in industry. The transcendental planar curves such as spirals and conics are inconsistent with the Computer Aided Design system. In CAD planar curves, which provide free-form mathematical description of shapes, are approximated to parametric curves and used as the basic building block. The planar curves such as circle, parabola, ellipse, hyperbola are employed in the research work presented here for shape expression, designing of mechanical accessories (tube benders, cutters, wrenches, clamp systems, inspection gauges), designing of railway and highway routes, construction of roller coaster and outline of the fonts in Computer Aided Design system. The cubic C-Bézier curve, cubic H-Bézier curve and parametric rational cubic curve are practiced to approximate these planar curves. These curves are in control point form and satisfy the properties of famous ordinary Bézier and spline curves. We establish that these curves, which are generalization of cubic Fergusons curves, cubic Bézier curves and cubic uniform B-spline curves are more efficient and closer to the control polygon than the ordinary Bézier and spline curves. The proposed approximation schemes are designed to control the geometric features of planar curves with the geometric constraints. The control points of the cubic C-Bézier curve, cubic H-Bézier curve and parametric rational cubic curve are evaluated by geometric approximation constraints. We use a number of optimization techniques to control the error of the proposed schemes and to provide a unique approximating curve for a given planar curve. The schemes in practice at present approximate planar curves in terms of control points and weights of rational quadratic Bézier curve. The main contribution of this thesis is that the proposed geometric approximation schemes are based on end tangents and curvatures of planar curves. Therefore, these approximation schemes do not need the rational quadratic Bézier representation of planar curves. Numerical experiments suggest that the presented approximation schemes of this thesis are simple, effective and feasible. The absolute errors for developed approximation schemes are less than the prevailing schemes. The smaller absolute error confirms the applicability and efficiency of the proposed methods.
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شیخ ظہور الحسن

شیخ ظہور الحسن
افسوس ہے کہ شیخ ظہورالحسن صاحب سابق ریوینیو سکریٹری حکومت اتر پردیش نے کراچی میں انتقال کیا، وہ اپنے محکمہ کے ماہر اور اس صوبے کے لائق ترین عہدہ داروں میں تھے، حکومت میں ان کا بڑا وقار تھا، عملاً بڑے مذہبی اور دیندار تھے، مذہبی مطالعہ وسیع تھا، دینی کاموں سے بڑی دلچسپی تھی، ترک وطن سے پہلے دارالمصنفین کی مجلس انتظامیہ کے رکن تھے، کئی سال ہوئے ریٹائرڈ ہوچکے تھے، ان کے لڑکے پاکستان میں بڑے عہدوں پر تھے، لیکن وہ خود ہندوستان ہی میں مقیم رہے، ۱۹۶۵؁ء کی جنگ سے کچھ پہلے لڑکوں سے ملنے کراچی گئے تھے، اسی دوران میں جنگ چھڑگئی، اس لیے لڑکوں کے اصرار سے وہاں مستقل قیام اختیار کرلیا، اور وہیں قلبی دورے میں وفات پائی، مسلمان عہدہ داروں میں ایسے لائق اور دیندار کم ہوتے ہیں، اﷲ تعالیٰ اس مرد مومن کی مغفرت فرمائے۔ (شاہ معین الدین ندوی، جنوری ۱۹۷۲ء)

Research on Learning Strategies in Arabic Language Education

The learning of Arabic language like any other foreign language contains four main aspects; reading, writing, speaking and understanding while listening.[i] This learning process can be enhanced if the most appropriate Learning Strategy is used. In this paper the most appropriate Learning Strategy of Arabic Language is suggested. The course outlines for Arabic language are thoroughly studied and several professors and experts of Arabic Language from Pakistan, Egypt, Saudi Arabia and Sudan are interviewed. The author, who himself has vast experience in teaching Arabic language, also had the opportunity to sit in the Arabic language classes to observe various strategies and methodologies adopted by different professors while teaching Arabic. In this paper the time spent on teaching Arabic to the students is also discussed. The appropriate size of the class room i.e, the number of students in Arabic language class also matters in improving the quality of Arabic among the students. The matter of teaching Arabic in Arabic only or in the native language of the students will also be touched in here. As the time has changed and the world is moving ahead on a fast pace, it seems necessary to apply the “Direct Method” while teaching Arabic or any foreign language.[ii] This paper will shed light on what is meant by “Direct Method”. The idea of making the student sit and memorize the dry rules of grammar has become obsolete. The idea of telling the student what part of the phrase is subject or predicate, or what is object and what is a noun or verb, may come later. The idea of memorizing the bulk of new vocabulary in the beginning can also be postponed. Hence a paradigm shift is needed here while talking about the Methodology of Teaching Arabic Language, under the heading of “Direct Method”.   [i]     Muhammad Abdul Khaliq, Professor of Arabic and co-author of 'al-Arabia baina Yadaik'. The author of this research paper had a personal interview with him on 21.03.2014, in the Institute of Arabic Language, King Saud University, Riyadh, Saudi Arabia. [ii]    This method is adopted roughly by some great scholars of Arabic language like Dr. V. Abdur Rahim who taught Arabic language for decades in the Islamic University of Madina, Kingdom of Saudi Arabia. The author was fortunate to meet with him many times and get benefitted from his experience. See for details: Abdurrahim, V. (1999), Arabic Course for English-Speaking Students, Leicester: UK Islamic Academy. See also: Abdullah, F. Ibrahim. (1999), Iqra Arabic Reader. Chicago: Iqra International Educational Foundation. Moreover see: Fawzan, Abdurrahman and others. (2004), Al-Arabia Baina Yadaik, Riyadh: Ministry of Education.

Formulation Characterization and Non-Invasive In-Vivo Evaluation of Dermocosmetic Emulgels Containing Various Fruit Extracts

Natural plant derived phytoconstituents with appreciable antioxidant activity have gained sufficient interest in cosmetic and dermatology for various clinical and aesthetic purposes. Various skin related disorders such as acne, inflammation, hyper-pigmentary disorders, sunburn, photoaging, melisma, and dry skin conditions are now becoming a point of focus to be treated with natural ingredients based topical formulations because of lower risk of adverse effects. In this research project bioactive, polyphenolic, antioxidant enriched fruit extracts from Diospyros Kaki (Persimmon), Manilkara Zapota (Cheeku) and Grewia Asiatica (phalsa) were evaluated for their anti-oxidant activities using DPPH free radical scavenging assay. Total phenolic and total flavonoid contents were estimated quantitatively. HPLC analysis was carried out for qualitative and quantitative analysis of polyphenols present in these extract samples. Finally, after phytochemical screening these fruit extract (10%) were loaded into topical emulgel formulations. The formulated emulgels (including active test formulations and respective control) were subjected to various stress storage conditions i.e. 8°C, 25°C, 40°C and 40°C+75%RH for a time period of 12 weeks, for evaluation of their physicochemical stability in terms of any change in colour, odour, consistency, pH, electrical conductivity, liquefaction, phase separation, and rheological attributes. Non-invasive In-vivo evaluation was performed using biophysical methods to evaluate the effects of active test formulations on various skin parameters such as melanin levels, erythema levels, moisture contents, sebum contents and facial skin elasticity. Surface evaluation of living skin was performed to estimate any variation in skin microreleif parameters such as scaliness, smoothness & wrinkles after treatment with active test formulation and respective control for a period of 90-days. Similarly, mean percent changes in porphyrin area, porphyrin count, porphyrin average intensity, number and area of fine and large facial skin pores was observed to elucidate and compare the effects of treatment with active test formulation to that of control. Sun protection factor was determined for formulated test and control formulations and their physical remanance on the skin was studied. Whole experimental work was carried out using single blinded, split face study design in three groups (Group-1, Group-2 and Group-3) with 13 male volunteers in each group. Results were analysed for their statistical significance using two way ANOVA and paired sample t-test. Antioxidant activities of Persimmon, Cheeku and Phalsa fruit extract were 79% ± 2.54, 82% ± 1.84 and 86% ± 1.03 respectively. Active test formulation loaded with Persimmon, Cheeku and Phalsa fruit extract and control exhibited optimum pH, electrical conductivity, and rheological attributes with good physical stability for a period of 3-months at accelerated stability storage conditions. The active test formulation EG-1, EG-2 and EG-3 produced significant (P≤0.05) while control exhibited insignificant (P>0.05) effects on melanin, erythema, sebum, elasticity, porphyrin area (%), porphyrin count and porphyrin average intensity, area and number of fine and large pores with respect to time. In case of moisture contents, active test formulations and control produced significant effects (P≤0.05), however the effect produced by active test formulation were much higher in magnitude. The active test formulation showed significant effects on skin scaliness, skin smoothness, and skin wrinkles while control produced insignificant effects with respect to time. Fruit extracts and active test formulations were shown to possess sufficient sunscreen activity and physical remanance on the skin as revealed by in-vitro spectrophotometric method and in-vivo biophysical technique. Moreover, the formulated active test emulgels were capable to quench both UVA and UVB radiations. Based on the findings of recent study it can be concluded that emulgels can be a stable, safe & cost-effective carrier system for transdermal delivery of antioxidant enriched extracts from Persimmon, Cheeku and Phalsa fruits. Moreover, formulated emulgels produced significant effects on various studied facial skin parameters by using non-invasive in-vivo biophysical techniques. Hence, it can safely be stated that these fruits can be used alone or in combination with other cosmetic ingredients to unveil their maximum beneficial potential for mankind.