Concyclic points
Template:Inline citations The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems (GIS), motion planning, and computer aided design (CAD).
In its most general form, the problem is, given a partition of the space into disjoint regions, determine the region where a query point lies. As an example application, each time you click a mouse to follow a link in a web browser, this problem must be solved in order to determine which area of the computer screen is under the mouse pointer. A simple special case is the point in polygon problem. In this case, we need to determine whether the point is inside, outside, or on the boundary of a single polygon.
In many applications, we need to determine the location of several different points with respect to the same partition of the space. To solve this problem efficiently, it is useful to build a data structure that, given a query point, quickly determines which region contains the query point.
Planar case
In the planar case, we are given a planar subdivision S, formed by multiple polygons called faces, and need to determine which face contains a query point. A brute force search of each face using the point-in-polygon algorithm is possible, but usually not feasible for subdivisions of high complexity. Several different approaches lead to optimal data structures, with O(n) storage space and O(log n) query time, where n is the total number of vertices in S. For simplicity, we assume that the planar subdivision is contained inside a square bounding box.
Slab decomposition
The simplest and earliest data structure to achieve O(log n) time was discovered by Dobkin and Lipton in 1976. It is based on subdividing S using vertical lines that pass through each vertex in S. The region between two consecutive vertical lines is called a slab. Notice that each slab is divided by non-intersecting line segments that completely cross the slab from left to right. The region between two consecutive segments inside a slab corresponds to a unique face of S. Therefore, we reduce our point location problem to two simpler problems:
- Given a subdivision of the plane into vertical slabs, determine which slab contains a given point.
- Given a slab subdivided into regions by non-intersecting segments that completely cross the slab from left to right, determine which region contains a given point.
The first problem can be solved by binary search on the x coordinate of the vertical lines in O(log n) time. The second problem can also be solved in O(log n) time by binary search. To see how, notice that, as the segments do not intersect and completely cross the slab, the segments can be sorted vertically inside each slab.
While this algorithm allows point location in logarithmic time and is easy to implement, the space required to build the slabs and the regions contained within the slabs can be as high as O(n²), since each slab can cross a significant fraction of the segments.
Several authors noticed that the segments that cross two adjacent slabs are mostly the same. Therefore, the size of the data structure could potentially be reduced by applying some kind of compression, where only the difference between two adjacent slabs is stored. Sarnak and Tarjan managed to use this idea to reduce the storage space to O(n), while maintaining the O(log n) query time. Unfortunately, the data structure becomes highly complex.
Monotone subdivisions
A (vertical) monotone chain is a path such that the y-coordinate never increases along the path. A simple polygon is (vertical) monotone if it is formed by two monotone chains, with the first and last vertices in common. It is possible to add some edges to a planar subdivision, in order to make all faces monotone, obtaining what is called a monotone subdivision. This process does not add any vertices to the subdivision (therefore, the size remains O(n)), and can be performed in O(n log n) time by plane sweep (it can also be performed in linear time, using polygon triangulation). Therefore, there is no loss of generality, if we restrict our data structure to the case of monotone subdivisions, as we do in this section.
The weakness of the slab decomposition is that the vertical lines create additional segments in the decomposition, making it difficult to achieve O(n) storage space. Edelsbrunner, Guibas, and Stolfi discovered an optimal data structure that only uses the edges in a monotone subdivision. The idea is to use vertical monotone chains, instead of using vertical lines to partition the subdivision.
Converting this general idea to an actual efficient data structure is not a simple task. First, we need to be able to compute a monotone chain that divides the subdivision into two halves of similar sizes. Second, since some edges may be contained in several monotone chains, we need to be careful to guarantee that the storage space is O(n). Third, testing whether a point is on the left or the right side of a monotone subdivision takes O(n) time if performed naively.
Details on how to solve the first two issues are beyond the scope of this article. We briefly mention how to address the third issue. Using binary search, we can test whether a point is to the left or right of a monotone chain in O(log n) time. As we need to perform another nested binary search through O(log n) chains to actually determine the point location, the query time is O(log² n). To achieve O(log n) query time, we need to use fractional cascading, keeping pointers between the edges of different monotone chains.
Triangulation refinement
A polygon with m vertices can be partitioned into m-2 triangles. There are numerous algorithms to triangulate a polygon efficiently, the fastest having O(n) worst case time. Therefore, we can decompose each polygon of our subdivision in triangles, and restrict our data structure to the case of subdivisions formed exclusively by triangles. Kirkpatrick gives a data structure for point location in triangulated subdivisions with O(n) storage space and O(log n) query time.
The general idea is to build a hierarchy of triangles. To perform a query, we start by finding the top-level triangle that contains the query point. Since the number of top-level triangles is bounded by a constant, this operation can be performed in O(1) time. Each triangle has pointers to the triangles it intersects in the next level of the hierarchy, and the number of pointers is also bounded by a constant. We proceed with the query by finding which triangle contains the query point level by level.
The data structure is built in the opposite order, that is, bottom-up. We start with the triangulated subdivision, and choose an independent set of vertices to be removed. After removing the vertices, we retriangulate the subdivision. Because the subdivision is formed by triangles, a greedy algorithm can find an independent set that contains a constant fraction of the vertices. Therefore, the number of removal steps is O(log n).
Trapezoidal decomposition
A randomized approach to this problem, and probably the most practical one, is based on trapezoidal decomposition, or trapezoidal map. A trapezoidal decomposition is obtained by shooting vertical bullets going both up and down from each vertex in the original subdivision. The bullets stop when they hit an edge, and form a new edge in the subdivision. This way, we obtain a subset of the slab decomposition, with only O(n) edges and vertices, since we only add two edges and two vertices for each vertex in the original subdivision.
It is not easy to see how to use a trapezoidal decomposition for point location, since a binary search similar to the one used in the slab decomposition can no longer be performed. Instead, we need to answer a query in the same fashion as the triangulation refinement approach, but the data structure is constructed top-down. Initially, we build a trapezoidal decomposition containing only the bounding box, and no internal vertex. Then, we add the segments from the subdivision, one by one, in random order, refining the trapezoidal decomposition. Using backwards analysis, we can show that the expected number of trapezoids created for each insertion is bounded by a constant.
We build a directed acyclic graph, where the vertices are the trapezoids that existed at some point in the refinement, and the directed edges connect the trapezoids obtained by subdivision. The expected depth of a search in this digraph, starting from the vertex corresponding to the bounding box, is O(log n).
Higher dimensions
There are no known general point location data structures with linear space and logarithmic query time for dimensions greater than 2. Therefore, we need to sacrifice either query time, or storage space, or restrict ourselves to some less general type of subdivision.
In three-dimensional space, it is possible to answer point location queries in O(log² n) using O(n log n) space. The general idea is to maintain several planar point location data structures, corresponding to the intersection of the subdivision with n parallel planes that contain each subdivision vertex. A naive use of this idea would increase the storage space to O(n²). In the same fashion as in the slab decomposition, the similarity between consecutive data structures can be exploited in order to reduce the storage space to O(n log n), but the query time increases to O(log² n).
In d-dimensional space, point location can be solved by recursively projecting the faces into a (d-1)-dimensional space. While the query time is O(log n), the storage space can be as high as . The high complexity of the d-dimensional data structures led to the study of special types of subdivision.
One important example is the case of arrangements of hyperplanes. An arrangement of n hyperplanes defines O(nd) cells, but point location can be performed in O(log n) time with O(nd) space by using Chazelle's hierarchical cuttings.
Another special type of subdivision is called rectilinear (or orthogonal) subdivision. In a rectilinear subdivision, all edges are parallel to one of the d orthogonal axis. In this case, point location can be answered in O(logd-1 n) time with O(n) space.
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
External links
- Point-Location Source Repository at Stony Brook University
- Point-Location Queries in CGAL, the Computational Geometry Algorithms Library