title: Three dimensional extension of Bresenham’s algorithm with Voronoi diagram

authors: Au, Chi Kit

subjects: voronoi diagram

abstract: Bresenham’s algorithm for plotting a two-dimensional line segment is elegant and efficient in its deployment of mid-point comparison and integer arithmetic. It is natural to investigate its three-dimensional extensions. In so doing, this paper uncovers the reason for little prior work. The concept of the mid-point in a unit interval generalizes to that of nearest neighbours involving a Voronoi diagram. Algorithmically, there are challenges. While a unit interval in two-dimension becomes a unit square in three-dimension, “squaring” the number of choices in Bresenham’s algorithm is shown to have difficulties. In this paper, the three-dimensional extension is based on the main idea of Bresenham’s algorithm of minimum distance between the line and the grid points. The structure of the Voronoi diagram is presented for grid points to which the line may be approximated. The deployment of integer arithmetic and symmetry for the three-dimensional extension of the algorithm to raise the computation efficiency are also investigated.

text: Three dimensional extension of Bresenham’s algorithm with Voronoi diagram Bresenham’s algorithm for plotting a two-dimensional line segment is elegant and efficient in its deployment of mid-point comparison and integer arithmetic. It is natural to investigate its three-dimensional extensions. In so doing, this paper uncovers the reason for little prior work. The concept of the mid-point in a unit interval generalizes to that of nearest neighbours involving a Voronoi diagram. Algorithmically, there are challenges. While a unit interval in two-dimension becomes a unit square in three-dimension, “squaring” the number of choices in Bresenham’s algorithm is shown to have difficulties. In this paper, the three-dimensional extension is based on the main idea of Bresenham’s algorithm of minimum distance between the line and the grid points. The structure of the Voronoi diagram is presented for grid points to which the line may be approximated. The deployment of integer arithmetic and symmetry for the three-dimensional extension of the algorithm to raise the computation efficiency are also investigated.

year: 2010

title: Identities in the Algebra of Partial Maps

authors: Jackson, Marcel

subjects: mathematics

abstract: We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable.

text: Identities in the Algebra of Partial Maps We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable.

year: 2006

title: Empowerment evaluation of Te Taiwhenua o Heretaunga Family Start: Improving service delivery

authors: Little, Gaylene Robina

subjects: Family Start

abstract: This thesis explores child maltreatment in New Zealand by considering service delivery of the Family Start programme at Te Taiwhenua o Heretaunga through an empowerment evaluation. The purpose of this research is to evaluate the service delivery process of the Family Start programme at this site to inform the organization and other Family Start key agents of possible improvements to service delivery so that the Family Start programme is best able to reduce the risk factors that are known to influence child maltreatment in New Zealand. The best possible service delivery by kaupapa providers supports sustainability and the continuity of service with Māori through continual funding. The two objectives of this research are to look at how well the Family Start programme is implemented at Te Taiwhenua o Heretaunga and to consider the cultural appropriateness of service delivery. Community psychology is the paradigm within which I position myself. I respect the values of community psychology that aim to improve the position of disadvantaged people through their participation in social change and community development. Te Taiwhenua o Heretaunga is a kaupapa Māori provider and I see an empowerment evaluation as a tool for internal evaluation to assist organizations who value self determination in their own practice. Both quantitative and qualitative data were collected and analyzed. There are two objectives to consider service delivery; firstly process aspects of service delivery are looked at and secondly the cultural relevance of service delivery to the population receiving the Family Start programme at this site. These are described as nine points about service delivery. The aim is to provide an empowerment evaluation for Te Taiwhenua o Heretaunga to inform, assist and improve service delivery of the Family Start programme in a culturally appropriate manner. The findings suggest better understanding is needed by Te Taiwhenua o Heretaunga Family Start staff, about the programme intentions, the use of tools such as Born to Learn/Ahuru Mōwai, individual family plans, service delivery levels, maintaining health records and ways to encourage collaboration between agencies. Te Taiwhenua o Heretaunga are shown in this research to be reaching the intended population for the Family Start programme, and service delivery appears to be culturally relevant to the clients on the programme at this time. The findings are limited by the fact that access to some information was restricted by Te Taiwhenua o Heretaunga Family Start management. Consequently, this research looked only at service delivery, and not the benefits of or barriers to the actual programme. The effectiveness of the programme in reducing child maltreatment is important but could not be measured in this research. Reducing child maltreatment is the main aim of the Family Start programme and would be measured through client outcomes. This research considered service delivery to see if the Family Start programme is offered optimally to assist the aim of reducing child maltreatment.

text: Empowerment evaluation of Te Taiwhenua o Heretaunga Family Start: Improving service delivery This thesis explores child maltreatment in New Zealand by considering service delivery of the Family Start programme at Te Taiwhenua o Heretaunga through an empowerment evaluation. The purpose of this research is to evaluate the service delivery process of the Family Start programme at this site to inform the organization and other Family Start key agents of possible improvements to service delivery so that the Family Start programme is best able to reduce the risk factors that are known to influence child maltreatment in New Zealand. The best possible service delivery by kaupapa providers supports sustainability and the continuity of service with Māori through continual funding. The two objectives of this research are to look at how well the Family Start programme is implemented at Te Taiwhenua o Heretaunga and to consider the cultural appropriateness of service delivery. Community psychology is the paradigm within which I position myself. I respect the values of community psychology that aim to improve the position of disadvantaged people through their participation in social change and community development. Te Taiwhenua o Heretaunga is a kaupapa Māori provider and I see an empowerment evaluation as a tool for internal evaluation to assist organizations who value self determination in their own practice. Both quantitative and qualitative data were collected and analyzed. There are two objectives to consider service delivery; firstly process aspects of service delivery are looked at and secondly the cultural relevance of service delivery to the population receiving the Family Start programme at this site. These are described as nine points about service delivery. The aim is to provide an empowerment evaluation for Te Taiwhenua o Heretaunga to inform, assist and improve service delivery of the Family Start programme in a culturally appropriate manner. The findings suggest better understanding is needed by Te Taiwhenua o Heretaunga Family Start staff, about the programme intentions, the use of tools such as Born to Learn/Ahuru Mōwai, individual family plans, service delivery levels, maintaining health records and ways to encourage collaboration between agencies. Te Taiwhenua o Heretaunga are shown in this research to be reaching the intended population for the Family Start programme, and service delivery appears to be culturally relevant to the clients on the programme at this time. The findings are limited by the fact that access to some information was restricted by Te Taiwhenua o Heretaunga Family Start management. Consequently, this research looked only at service delivery, and not the benefits of or barriers to the actual programme. The effectiveness of the programme in reducing child maltreatment is important but could not be measured in this research. Reducing child maltreatment is the main aim of the Family Start programme and would be measured through client outcomes. This research considered service delivery to see if the Family Start programme is offered optimally to assist the aim of reducing child maltreatment.

year: 2010