I want to make a polygon selection based on a specific number of points.

For example, I want to select those polygons that contain more than 30 points. I don't want to select every polygon that contains a point.

I tried to do something with ArcGis's data reviewer and I checked spatial join but I didn't find anything.

Ok, here is a way to do it that isn't too complicated.

Create a new short integer field in your points layer.

Assign each feature a value of 1.

Spatial join your polygon layer to your points layer.

For your options, choose that *each polygon will be given a summary of the numeric attributes… * option and summarize the values by **Sum**.

In your new layer, look for those who have the summed value of 30 or more.

Then you can export those selected features into a new layer. To select features from your original layer, do a *select by location* query against this exported layer, and you should have it.

## Find Hot Spots

This functionality is currently only supported in Map Viewer Classic (formerly known as Map Viewer ). It will be available in a future release of the new Map Viewer (formerly known as Map Viewer Beta ).

The Find Hot Spots tool will determine if there is any statistically significant clustering in the spatial pattern of your data.

## Find polygons that contain a specific number of points - Geographic Information Systems

One of the major problems when developing object detection algorithms is the lack of labeled data for training and testing many object classes. The goal of this database is to provide a large set of images of natural scenes (principally office and street scenes), together with manual segmentations/labelings of many types of objects, so that it becomes easier to work on general multi-object detection algorithms.

For getting the database and Matlab code follow the next link: Download Database

If you find this dataset usefull, help us to build a larger dataset of annotated images (which will be made available very soon) by using the web annotation tool written by Bryan C. Russell at MIT:

**Overview of the database content**

Here there are some of the characteristics of the database:

The following images show some example of annotated frames (static frames and sequences):

Each labeled image in the database is associated with an annotation ASCII file. This is an example of one annotation file:

# - List of Polygons

size

makePolygon

mouse makePolygon

deskFrontal makePolygon

poster makePolygon

keyboard makePolygon

cpu makePolygon

screenFrontal view

makePolygon

light

The next table is a list of all the object labels used in the annotations. Some of the labels correspond to parts of objects. The objects denoted with a (*) are interesting object for training detectors (interesting means that there are a reasonable number of annotated instances and some control for the variability of the object appearance):

'apple' (*) 'bicycle' 'bicycleSide' 'bookshelf' 'bookshelfFrontal' (*) 'bookshelfPart' 'bookshelfSide' 'bookshelfWhole' 'bottle' (*) 'building' 'buildingPart' 'buildingWhole' 'can' (*) 'car' (*) 'carFrontal' (*) 'carPart' 'carSide' (*) 'cd' (*) 'chair' 'chairPart' 'chairWhole' (*) 'coffeemachine' 'coffeemachinePart' | 'coffeemachineWhole' (*) 'cog' 'cpu' (*) 'desk' 'deskFrontal' (*) 'deskPark' 'deskPart' 'deskWhole' 'donotenterSign' (*) 'door' 'doorFrontal' (*) 'doorSide' 'filecabinet' 'firehydrant' (*) 'freezer' 'frontalFace' (*) 'frontalWindow' 'head' (*) 'keyboard' (*) 'keyboardPart' 'keyboardRotated' 'light' (*) 'mouse' (*) | 'mousepad' (*) 'mug' (*) 'onewaySign' (*) 'paperCup' (*) 'parkingMeter' (*) 'person' 'personSitting' 'personStanding' 'personWalking' (*) 'poster' (*) 'posterClutter' 'pot' (*) 'printer' 'projector' 'screen' 'screenFrontal' (*) 'screenPart' 'screenWhole' (*) 'shelves' 'sink' 'sky' 'sofa' | 'sofaPart' 'sofaWhole' 'speaker' (*) 'steps' 'stopSign' (*) 'street' 'streetSign' 'streetlight' 'tableLamp' (*) 'telephone' (*) 'torso' 'trafficlight' (*) 'trafficlightSide' 'trash' 'trashWhole' (*) 'tree' 'treePart' 'treeWhole' 'wallClock' 'watercooler' 'window' |

Here there is a histogram of counts for each labeled object (or parts of objects). The vertical axis is the number of labeled instances (resolution varies).

The frames are also labeled according to scene type (office, corridor, street, conference room, etc.)

**Structure of the annotation files**

This is an example of one annotation file:

# - List of Polygons

size

makePolygon

mouse makePolygon

deskFrontal makePolygon

poster makePolygon

keyboard makePolygon

cpu makePolygon

screenFrontal view

makePolygon

light

One object is described by a polygon:

The field "labels" allows to add additional information to describe an object. For instance, in the case of a "face", we might want to add information like the gender or the identity. The labels can be arbitrary:

We can then query to find objects with specific labels:

keys = queryDB(DB, 'findObject', 'frontalFace', 'findLabel', 'gender=male')

**MATLAB tools for handling the annotation files**

We have developed some MATLAB tools for using the database. The first set of function allows to read and create annotation files. The second set of function provides higher-level functions for indexing the annotations.

**Reading and plotting images**

There are four basic functions for reading, writing and plotting the annotation files:

All these four functions describe the polygons on an image using a struct array:

pf(:).class

pf(:).polygon

pf(:).vertices

pf(:).view

pf(:).labels

**Queries to the database**

There are some basic MATLAB tools to make queries to the database in order to locate the frames that contain specific objects or scenes.

1) First you have to create the database.

DB = makeDB('C:/images', 'C:/anno', 'C:/places')

The arguments are the directories in which the images, object annotations and place labels are stored.

The result of this function is the struct 'DB' which is an index for the database. This operation will take some time but you only have to do it once. Once is done, you can store the struct DB somewhere for future use.

>> keys = queryDB(DB, 'findObject', 'screenFrontal')

1x560 struct array with fields:

frame

objects

'keys' are pointers to frames and objects within each frame. For instance:

This indicates that the first image that contains a 'screenFrontal' is frame number 318, and the object is number 3 in the annotations. Therefore:

className: 'screenFrontal'

vertices: [2x4 double]

center: [527.4829 261.4052]

area: 139415

bbox: [4x1 double]

view: [2x1 double]

You can visualize some of the images with:

Some other query examples:

>> keys = queryDB(DB, 'findObject', 'coffeemachineWhole', 'findObject', '

>> keys = queryDB(DB, 'findObject', 'car*')

>> showImages(DB, [keys(1:10).frame])

3) Searching points of views

For some objects, we have also labeled the point of view. The labeling of the point of view is done by adding one line into the annotation file, just after the object polygon. For instance:

Here there are some examples of objects and the views used:

It is possible to find objects in the database using the point of view as a query argument:

>> keys = queryDB(DB, 'findObject', 'car*', 'findAzimuth', 90)

This returns frames that contain views of backs of cars (and other objects too):

Using folder names in the query is useful to create **training and test datasets that are independent**. Here we give some examples of useful queries:

keys = queryDB(DB, 'findFolder', 'seq')

keys = queryDB(DB, 'findFolder', 'static')

Get all images retrieved from the web:

keys = queryDB(DB, 'findFolder', 'web')

Get all images from building 200 (old AI-Lab building):

keys = queryDB(DB, 'findFolder', 'bldg200')

Get all images from Stata center (new CSAIL building):

keys = queryDB(DB, 'findFolder', 'stata')

Queries can be combined to locate instances of an object within a set of images:

keys1 = queryDB(DB, 'findObject', 'frontalScreen', 'findFolder', 'bldg200')

keys2 = queryDB(DB, 'findObject', 'frontalScreen', 'findFolder', 'stata')

Now, keys1 and keys2 are pointers to images containing "screens" taken in different buildings and therefore, provide a possible split in training and test sets.

keys = queryDB(DB, 'findLocation', '400_fl_608')

For getting the database and Matlab code follow the next link: Download Database

**Links to object detection and scene recognition code**

A. Torralba, K. P. Murphy, W. T. Freeman and M. A. Rubin.

Proceedings of the IEEE International Conference on Computer Vision, ICCV 2003, vol.1, p.273. Nice, France.

Code and demos: Context-based vision systemfor place and object recognition

**Related papers using this dataset**

A. Torralba, K. P. Murphy and W. T. Freeman. (2004). Sharing features: efficient boosting procedures for multiclass object detection.

Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition(CVPR). pp 762- 769. Also, see extended paper (MIT AI Lab Memo AIM-2004-008)A. Torralba, K. P. Murphy and W. T. Freeman (2004). Contextual Models for Object Detection using Boosted Random Fields. MIT AI Lab Memo AIM -2004-008, April 14.

K. P. Murphy, A. Torralba and W. T. Freeman (2003). Using the forest to see the trees: a graphical model relating features, objects and scenes .

Adv. in Neural Information Processing Systems16 (NIPS), Vancouver, BC, MIT Press.A. Torralba, K. P. Murphy, W. T. Freeman and M. A. Rubin (2003). Context-based vision system for place and object recognition,

IEEE Intl. Conference on Computer Vision(ICCV), Nice, France, October.

If you have comments about the dataset that you think can be useful for others to know, send us an email and we can post your comments here.

The database is open for contributions both in code and in annotations. We can add links to your contributions (send an email to any of us: Antonio Torralba, Kevin P. Murphy, William T. Freeman) . The goal is to have a database that grows beyond what is possible to do for a unique lab.

Egon Pasztor made many contribution in the early stages of the database. We also want to give thanks to the flight delays and specially to the bad television programs who motivated us very much into annotating more images every day.

## Diet and Behavior

White-tailed deer are herbivores, leisurely grazing on most available plant foods. Their stomachs allow them to digest a varied diet, including leaves, twigs, fruits and nuts, grass, corn, alfalfa, and even lichens and other fungi. Occasionally venturing out in the daylight hours, white-tailed deer are primarily nocturnal or crepuscular, browsing mainly at dawn and dusk.

In the wild, white-tails, particularly the young, are preyed upon by bobcats, mountain lions, and coyotes. They use speed and agility to outrun predators, sprinting up to 30 miles per hour and leaping as high as 10 feet and as far as 30 feet in a single bound.

Although previously depleted by unrestricted hunting in the United States, strict game-management measures have helped restore the white-tailed deer population.

## Clarity, Treatments, and Durability

Emerald has a Mohs hardness of 7.5 to 8, which is normally a very good hardness for jewelry use. However, most emeralds contain numerous inclusions or surface-reaching fractures. These can weaken the gem, cause it to be brittle, and make it subject to breakage.

These are expected characteristics of emerald. It is rare to find an emerald that does not have inclusions and surface-reaching fractures that can be seen with the unaided eye. Under low magnification, most emeralds are said to have a "garden" of inclusions.

To improve appearance, most cut emeralds are treated with oils, waxes, polymers, or other substances that enter the fractures and make them less obvious. Although these treatments might improve appearance, they often do not improve the durability of the gem and they may discolor or deteriorate over time.

With that information, emerald should be considered a fragile stone that is best worn as a ring stone on special occasions rather than daily. Emerald is better suited for earrings and pendants that are usually subjected to less impact and abrasion than rings and bracelets. Settings that protect the stone are much safer than those that present the stone to impact and abrasion.

Cleaning emeralds should be done carefully. Steam and ultrasonic cleaning can remove oils and other fracture-filling treatments. A light washing in warm water with a mild soap is safer for cleaning and should be done only when needed.

**Emerald imports:** This graph illustrates the popularity of emeralds in the United States. The pie represents all colored stones imported into the United States during 2015 on the basis of dollar value. As a single gem variety, emerald holds the biggest share of the pie. More dollars' worth of emeralds were imported than any other colored stone. More dollars' worth of emeralds were imported than ruby and sapphire combined. Data from the USGS Minerals Yearbook, March 2018. [1]

**Gemstone import value:** This chart shows the quantity and value of diamond, emerald, ruby, sapphire, and other colored stones imported into the United States during 2015. This chart shows that, on the basis of cut but unset value, emerald is the most important gemstone import for the United States after diamond. It also has an average per-carat price that is much higher than ruby and sapphire. These amounts are approximately equal to consumption because the amount of domestic production was just several million dollars total. Data from the USGS Minerals Yearbook, March 2018. [1]

## Find polygons that contain a specific number of points - Geographic Information Systems

A **transaction** is a single logical unit of work which accesses and possibly modifies the contents of a database. Transactions access data using read and write operations.

In order to maintain consistency in a database, before and after the transaction, certain properties are followed. These are called **ACID** properties.

**Atomicity**

By this, we mean that either the entire transaction takes place at once or doesn’t happen at all. There is no midway i.e. transactions do not occur partially. Each transaction is considered as one unit and either runs to completion or is not executed at all. It involves the following two operations.

—**Abort**: If a transaction aborts, changes made to database are not visible.

—**Commit**: If a transaction commits, changes made are visible.

Atomicity is also known as the ‘All or nothing rule’.

Consider the following transaction **T** consisting of **T1** and **T2**: Transfer of 100 from account **X** to account **Y**.

If the transaction fails after completion of **T1** but before completion of **T2**.( say, after **write(X)** but before **write(Y)**), then amount has been deducted from **X** but not added to **Y**. This results in an inconsistent database state. Therefore, the transaction must be executed in entirety in order to ensure correctness of database state.

**Consistency**

This means that integrity constraints must be maintained so that the database is consistent before and after the transaction. It refers to the correctness of a database. Referring to the example above,

The total amount before and after the transaction must be maintained.

Total **before T** occurs = **500 + 200 = 700**.

Total **after T occurs** = **400 + 300 = 700**.

Therefore, database is **consistent**. Inconsistency occurs in case **T1** completes but **T2** fails. As a result T is incomplete.

**Isolation**

This property ensures that multiple transactions can occur concurrently without leading to the inconsistency of database state. Transactions occur independently without interference. Changes occurring in a particular transaction will not be visible to any other transaction until that particular change in that transaction is written to memory or has been committed. This property ensures that the execution of transactions concurrently will result in a state that is equivalent to a state achieved these were executed serially in some order.

Let **X**= 500, **Y** = 500.

Consider two transactions **T** and **T”.**

Suppose **T** has been executed till **Read (Y)** and then **T’’** starts. As a result , interleaving of operations takes place due to which **T’’** reads correct value of **X** but incorrect value of **Y** and sum computed by **T’’: (X+Y = 50, 000+500=50, 500)**

is thus not consistent with the sum at end of transaction: **T: (X+Y = 50, 000 + 450 = 50, 450)**.

This results in database inconsistency, due to a loss of 50 units. Hence, transactions must take place in isolation and changes should be visible only after they have been made to the main memory.

**Durability:**

This property ensures that once the transaction has completed execution, the updates and modifications to the database are stored in and written to disk and they persist even if a system failure occurs. These updates now become permanent and are stored in non-volatile memory. The effects of the transaction, thus, are never lost.

The **ACID** properties, in totality, provide a mechanism to ensure correctness and consistency of a database in a way such that each transaction is a group of operations that acts a single unit, produces consistent results, acts in isolation from other operations and updates that it makes are durably stored.

This article is contributed by **Avneet Kaur**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to [email protected] See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Attention reader! Don&rsquot stop learning now. Learn all **GATE CS concepts with Free Live Classes** on our youtube channel.

## Find polygons that contain a specific number of points - Geographic Information Systems

We are given an array of n points in the plane, and the problem is to find out the closest pair of points in the array. This problem arises in a number of applications. For example, in air-traffic control, you may want to monitor planes that come too close together, since this may indicate a possible collision. Recall the following formula for distance between two points p and q.

The Brute force solution is O(n^2), compute the distance between each pair and return the smallest. We can calculate the smallest distance in O(nLogn) time using Divide and Conquer strategy. In this post, a O(n x (Logn)^2) approach is discussed. We will be discussing a O(nLogn) approach in a separate post.**Algorithm**

Following are the detailed steps of a O(n (Logn)^2) algorithm. *Input:* An array of n points *P[]* *Output:* The smallest distance between two points in the given array.

As a pre-processing step, the input array is sorted according to x coordinates.**1)** Find the middle point in the sorted array, we can take *P[n/2]* as middle point. **2)** Divide the given array in two halves. The first subarray contains points from P[0] to P[n/2]. The second subarray contains points from P[n/2+1] to P[n-1].**3)** Recursively find the smallest distances in both subarrays. Let the distances be dl and dr. Find the minimum of dl and dr. Let the minimum be d.

**4)** From the above 3 steps, we have an upper bound d of minimum distance. Now we need to consider the pairs such that one point in pair is from the left half and the other is from the right half. Consider the vertical line passing through P[n/2] and find all points whose x coordinate is closer than d to the middle vertical line. Build an array strip[] of all such points.

**5)** Sort the array strip[] according to y coordinates. This step is O(nLogn). It can be optimized to O(n) by recursively sorting and merging. **6)** Find the smallest distance in strip[]. This is tricky. From the first look, it seems to be a O(n^2) step, but it is actually O(n). It can be proved geometrically that for every point in the strip, we only need to check at most 7 points after it (note that strip is sorted according to Y coordinate). See this for more analysis.**7)** Finally return the minimum of d and distance calculated in the above step (step 6)**Implementation**

Following is the implementation of the above algorithm.

## Isolines

Topographic maps use a wide variety of symbols to represent human and physical features. Among the most striking are the topo maps' display of the topography or terrain of the area.

Contour lines are used to represent elevation by connecting points of equal elevation. These imaginary lines do a nice job of representing the terrain. As with all isolines, when contour lines lie close together, they represent a steep slope lines far apart represent a gradual slope.

## Find polygons that contain a specific number of points - Geographic Information Systems

**This activity was developed by a student or students at Mainland High School which is located in Daytona Beach, FL. It is still a "work in progress" with editing and improvements yet to come.**

Click on a letter to jump to that section of the page

**Absolute value**

A number's distance from zero (0) on a number line. The absolute value of both 4, written |4|, and negative 4, written |-4|, equals 4.

** Acute Angle**

An angle with a measure of less than 90 degrees.

**Additive identity**

The number (0), that is, adding 0 does not change a number's vale (e.g., 5 + 0 = 5).

**Additive inverse property**

A number and its additive inverse have a sum of zero (0) (e.g., in the equation 3 + -3 = 0, 3 and -3 are additive inverses of each other).

**Algebraic equation**

A mathematical sequence in which two expressions are connected by an equality symbol.

**Algebraic expression**

A mathematical sentence in which two expressions are connected by an equality symbol.

**Algebraic order of operations**

An expression containing numbers and variables (e.g., 7x), and operations that involve numbers and variables (e.g., 2*x* + *y* or 3*a* - 4). Algebraic expressions do not contain equality or inequality symbols.

**Algebraic rule**

A mathematical expression that contains variables and describes a pattern or relationship.

The shape made by two rays extending from a common end point, the vertex. Measures of angles are described using the degree system.

The inside region of a two-dimensional figure measured in square units (e.g., a rectangle with sides of 4 units by 6 units contains 24 square units or has an area 24 square units).

**Associative property**

The way in which three or more numbers are grouped for addition or multiplication does not change their sum or product (e.g., 2 + 3 = 3 +2 or 4 x 7 = 7 x 4).

The horizontal and vertical number lines used in a rectangular graph or coordinate grid system.

The line or plane upon which a figure is thought of as a resting.

A zigzag on the line of the *x*- or *y*-axis in a line or a bar graph indicating that the data being displayed does not include all of the values that exist on the number line being used. Also called a *Squiggle.*

The amount of space that can be filled. Both capacity and volume are used to measure three-dimensional spaces how ever, capacity usually refers to fluids, whereas volume usually refers to solids.

**Circumference**

The perimeter of a circle is called its circumference.

**Closed figure**

A two-dimensional figure whose beginning and ending points meet, such that the plane in which the figure lies is divided into two parts---the part inside the figure and the part outside the figure (e.g., circles, squares, rectangles).

**Commutative property**

The order in which two numbers are added or multiplied does not change their sum or product (e.g., 2 + 3 = 3 +2 or 4 x 7 = 7 x 4).

**Complementary angles**

Two angles, the sum of which is exactly 90 degrees.

A whole number that has no more than two factors.

**Concrete representations of numbers**

Having a definite for or relating to an actual thing.

Figures or objects that are the same shape and the same size.

**Coordinate grid or system**

A network of evenly spaced, parallel horizontal and vertical lines especially designed for locating points, displaying data, or drawing maps.

Numbers that correspond to points on a graph in the form (*x , y).*

**Customary units**

The units of measure developed and used in the United States. Customary units for length are inches, feet, yards and miles. Customary units for volume are cubic inches, cubic feet, and cubic yards. Customary units for capacity or fluid ounces, cups, pints, quarts, and gallons.

**Data displays**

Different ways of displaying data in tables, charts, or graphs, including pictographs, circle graphs, single, double, or triple bar and line graphs, histograms, stem-and-leaf plots, and scatter plots.

**Decimal number**

Any number written with a decimal point in the number. A decimal number falls between two whole numbers (e.g., 1.5 falls between 1 and 2). Decimal numbers smaller than 1 are sometimes called decimal fractions (e.g., five-tenths is written 0.5).

A line segment from any point on the circle passing through the center to another point on the circle.

**Direct measure**

Obtaining the measure of an object by using measuring devices, either standard devices of the customary or metric systems, or nonstandard devices such as a paper clip or pencil.

**Distributive property**

For any real numbers *a, b,* and *x, x*(*a + b) = ax + bx.*

**Effects of operations**

The results of applying an operation to given numbers (e.g., adding two whole numbers results in a number greater than or equal to the original numbers).

An increase in size in all directions by a uniform amount.

A mathematical sentence (e.g., 2*x* = 10) that equates one expression (2*x*) to another expression (10).

**Equivalent expressions**

Expressions that have the same value but are represented in a different format using the properties of numbers [e.g., *ax + bx = (a +b)x].*

**Equivalent forms of a number**

Expressions that have the same value but are represented in a different format using the properties of numbers [e.g., *ax + bx = (a + b)x].*

The use of rounding and/or other strategies to determine a reasonably accurate approximation, without calculating an exact answer.

**Evaluate an expression**

Substitute numbers for the variables and follow the operation symbols to find the numerical value of the expression.

**Explain in words**

Directions requesting a written description of the procedures for finding the solution to the problem presented.

**Exponent (exponential form)**

The number of times the base occurs as a factor. For example, 2^3 is the exponential form of 2x2x2. The numeral two (2) is called the base, and the numeral three (3) is called the exponent.

A collection of numbers, symbols, and/or operation signs that stands for a number.

To estimate or infer a value or quantity beyond the known range.

One of the plane surfaces bounding a three-dimensional figure (a side).

A number or expression that divides exactly another number (e.g., 1,2,3,4,5,10, and 20 are factors of 20).

**Finite graph**

A graph having definable limits.

A transformation that produces the mirror image of a geometric figure. Also called a **reflection.**

Any part of a whole is called a fraction (e.g., one-half written in fractional form is 1/2.

The relationship between two sets (e.g., sets of numbers) in which each element of one set has one assigned element in the other set. See *Pattern*

**Function table**

A table of *x-* and *y-*values (ordered pairs) that represents the function, pattern, relationship, or sequence between the two variables.

A network of evenly spaced, parallel horizontal and vertical lines.

A line segment extending from the vertex or apex of a figure to its base and forming a right angle with the base or basal plane.

A proposition or supposition developed to provide a basis for further investigation or research.

**Indirect measure**

Obtaining the measurement of an object through the known measure of another object.

A sentence that states one expression is greater than or equal to, less than, less than or equal to another expression (e.g., a does not = 5 or x < 7 ).

The value of a variable when all other variables in the equation equal zero (0). On a graph, the values where a function crosses the axes.

**Intersection**

The point at which two lines meet.

**Inverse operation**

An action that cancels a previously applied action. For example, subtraction is the inverse operation of addition.

**Irrational number**

A real number that can not be expressed as a ratio of two numbers (e.g., 20=2(*w*+4) + 2*w* and *y* = 3*x* + 4).

**Labels (for a graph)**

The titles given to a graph, the axes of a graph, or to the scales on the axes of a graph.

A one-dimensional measure that is the measurable property of line segments,

The chance that something is likely to happen. **See Probability.**

A straight line that is endless in length.

**Line segment**

A portion of a line that has a defined beginning and end (e.g., the line segment AB is between point A and point B).

**Linear equation**

An algebraic equation in which the variable quantify or quantities are in the first power and the graph is a straight line (e.g., 20 = 2(w + 4) + 2*w* and *y* = 3*x* + 4).

The arithmetic average of a set of ordered numbers where half of the numbers are above the median and half are below it.

The middle point of a set of ordered numbers where half are below it.

**Metric units**

The units of measure developed in Europe and used in most of the world. Like the decimal system, the metric system uses the base 10. Metric units for length are millimeters, centimeters, meters, kilometers. Metric units for weight are milligrams, grams, and kilograms. Metric units for volume are cubic millimeters, cubic centimeters, and cubic meters. Metric units for capacity are milliliters, centiliters, liters, and kiloliters.

**Midpoint of a line segment**

The point on a line segment that divides it into two equal parts.

The score or data point found most often in a set of numbers.

The numbers that result from multiplying a given number by the set of whole numbers (e.g., the multiples of 15 are 0, 15, 30, 45, 60, 75, etc. ).

**Multiplicative identity**

The number one (1), that is, multiplying by 1 does not change a number's value (e.g., 5 x 1= 5).

**Multiplicative inverse** (reciprocal)

Any two numbers with a product of 1. (e.g., 4 and 1/4).

**Natural numbers** (counting numbers)

**Negative exponent**

Used in scientific notation to designate a number smaller than one (1) (e.g., 3.45 x 10^-2 equals0.0345).

**Nonstandard units of measure**

Units such as blocks, paper clips, crayons, or pencils that can be used to obtain a measure.

** Number line**

A line on which numbers can be written or visualized.

**Obtuse angle**

An angle with a measure of more than 90 degrees but less than 180 degrees.

The ratio of one event occuring to it not occuring.

Any mathmatical process, such as addition, subtraction, multiplication, division, exponents, or square roots.

**Operational shortcut**

A method having fewer arithmetic calculations.

**Ordered pair**

The location of a single point on a rectangular coordinate system where the digits represent the position relative to the *x*-axis and *y*-axis [e.g., (*x,y*) or (3,4)]

**Organize data**

To arrange data in a display that is meaningful and that assists in the interpretation of the data. See ** Data displays**.

**Parallel lines**

Two lines in the same plane that never meet. Also, lines with equal slopes.

**Pattern** **(relationship)**

A predictable or prescribed sequence of numbers, objects, etc. Patterns and relationships may be described or presented using munipulatives, tables, graphics (pictures or drwings), or algebraic rules (functions). Also called a **Relation.**

A special-case ratio in which the second term is always 100. The ratio is written as a whole number followed by a percent sign (e.g., 25% means the ratio of 25 to 100).

The length of the boundary around a figure.

**Perpendicular**

The symbol designating the ratio of the circumference of a circle to its diameter, represented as either 3.17 or 22/7.

**Place value**

The position of a single digit in a whole number or decimal number containing one or more digits.

**Planar cross section**

The intersection of a plane and a three-dimensional figure.

An undefined, two-dimensional (no depth) geometric surface that has no boundries specified. A plane is determined by defining points or lines exisiting on the plane.

**Plane figure**

A two-dimensional figure that lies entirely within a single plane.

A location in space that has no length or width.

A closed plane figure whose sides are straight lines and do not cross.

**Prime number**

Any whole number with only two factors, 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).

A three-dimensional figure (polyhedron) with congruent polygonal bases and lateral faces that are all parallelograms.

** Probability**

The likelihood of an event happening. An impossible event has a probability of zero. An event that will occur with absolute certainty is assigned a probability of one. Every event that is neither certain nor impossible has a probability that is between zero and one, and is obtained by dividing the number of favorable outcomes of an event by the total number of possible outcomes.

**Probability, empirical**

The likelihood of an event happening that is based on experience and observation rather than on theory.

**Probability, theoretical**

The likelihood of an event happening that is based on theory rather than on experience and observation.

A set of steps that demonstrates the truth of a given statement. Each step can be justified with a reason, such as a given, a definition, an axiom, or a previously proven property.

**Pythagorean theorem**

The square of the hypotenuse (*c*) of a right triangle is equal to the sum of the square of the legs (*a* and *b*), as shown in the equation *c^*2=*a*^2+*b*^2.

Any of the four regions formed by the axes in a rectangular coordinate system.

**Radical sign**

The symbol used before a number to show that the number is radicand.

A number that appears with a radical sign.

A line segment exrtending from the center of a circle or sphere to a point on the circle or sphere.

**Range of a set of numbers**

The difference between the highest (H) and the lowest (L) value in a set of data sometimes calculated as H - L + 1.

Calculations involving rates, distances, and time intervals, based on the distance, rate, time formula (D = *r t*).

The compression of two quantities (e.g., the ratio of *a* and *b* is a/b, where b doesn't equal zero).

**Rational number**

A real number that can be expressed as a ratio of two integers.

A portion of a line that begins at a point and goes on forever in one direction.

**Real numbers**

All rational and irrational numbers.

** Reflection**

**Reflexive axiom of equality**

A number or expression is equal to itself (e.g.,*ab* = *ab*).

**Regular polygon**

A polygon that is both quilateral and quiangular.

** Relation (relationship)**

**Relative size**

The size of one number in comparison to the size of another number or numbers.

**Right angle**

An angle whose measure is exactly 90 degrees.

**Right circular cylinder**

A cylinder in which the bases are parallel circles perpendicular to the side of the cylinder.

**Right triangle geometry**

Finding the measures of missing sides or angles of a right triangle when given the measures of other sides or angles. See **Pythagorean theorem.**

The change in *y* going from one point of *y* to another (the horizontal change on the graph.)

A transformation of a figure by turning it about a center point or axis. The amount of rotation is usually expressed in the number of degrees (e.g., a 90 degree rotation). Also called a **Turn.**

A mathmatical expression that describes a pattern or relationship, or a written description of the pattern or relationship.

The change in *x* going from one point of *y* to another (the horizontal change on the graph).

**Scale model**

A model or drwaing based on a ratio of the dimensions for the model and the actual object it represents (e.g., a map).

The numeric values assigned to the axes of a graph.

**Scatter plot**

A graph of data points, usually from an experiment, that is used to observe the relationship between two variables.

**Scientific notation**

A shorthand method of writing very large or very small numbers using exponents in which a number is expressed as the product of a power of 10 and a number that is greater than or equal to one (1) and less than 10 (e.g.,7.59 x 10^5=759,000). It is based on the idea that is easier to read exponents than it is to count zeros. If a number is already a power of 10, it is simply written 10^27 instead of 1 x 10^27.

An ordered list with either a constant difference (arithmetic) or a constant ratio (geomtric).

The edge of a geometric figure (e.g., a triangle has three sides).

**Similar figures**

Two figures that are the same shape, have corresponding, congruent angles, and having coorisponding sides that are proportional in length.

Figures that are the same shape are similar they are not necessarily the same size or in the same position.

To move along in constant contract with the surface in a vertical, horizontal, or diagonal direction. Also called a **Translation.**

The incline of a line, defined by the ratio of the change in units on the vertical axis to the change in one unit on the horizontal axis.

**Solid figures**

Three-dimensional figures that completely enclose a portion of space (e.g., a reatangularsolid, cube, sphere, right circular cylindar, right ciscular cone, and regular square pyramid).

**Spatial relationships**

Relationships of figures existing or happening in space.

**Square root**

A positive real number that can be multiplied by itself to produce a given number (e.g., the square root of 144 is 12, or =12).

**Standard units of measure**

The measurement of an object by using accepted measuring devices and units of the customary or metric system.

**Straight angle**

An angle whose measure is exactly 180 degrees.

**Supplementary angles**

Two angles, the sum of which is exactly 180 degrees.

**Surface area of a geometric solid**

The sum of the area of the faces of the figure that create the geometric solid.

**Symbolic expression**

A symbol or set of symbols expressing a mathmatical quantity or operation (e.g., 2*x* is equal to two times *x*).

**Symbolic representations of numbers**

Being expressed by symbols (e.g., circles shaded to represent 1/4, or variables used to represent quantities).

When a line can be drawn through the center of a figure such that the two halves are congruent.

**Systems of equations**

A group of two or more equations that share variables. The solution to a system of equations is an ordered number set that makes all of the equations true.

**Tessellation**

A covering of a plane with congruent copies of the same pattern with no holes and no overlaps, like floor tiles.

** Transformation**

An operation on a geometric figure by which another image is created. Common transformations include flips, slides, and turns.

** Transitive property**

When the first element has a particular relationship to a third element that in turn has the same relationship to a third element, the first has this same relationship to the third element (e.g., if *a = b* and *b = c,* then *a = c*)*.* Identity and equality are transitive relationships.

** Translation**

** Tree diagram**

A diagram in which all the possible outcomes of a given event are displayed.

**Unorganized data**

Data that are presented in a random manner.

Any symbol that could represent a number.

The common endpoint from which two rays begin (i.e., the vertex of an angle) or the point where two lines intersect the point on a triangle or pyramid opposite to and farthest from the base.

** Vertical angles**

The oppisite angles formed when two lines intersect.

The amount of space occupied in three dimensions and expressed in cubic units. Both capacity and volume are used to measure empty spaces however, capacity usually refers to fluids, whereas volume usually refers to solids.

Measures that represent the force that attracts an object to the center of Earth. In the customary system, the basic unit of weight is the pound.

** Whole numbers**

The value of *x* on a graph when *y* is zero (0). The *x*-axis is the horizontal number line on a rectangular coordinate system.

The value of *y* on a graph when *x* is zero (0). The *y*-axis is the vertical number line on a rectangular coordinate system.

## Clustering Methods

Clustering methods can be classified into the following categories &minus

- Partitioning Method
- Hierarchical Method
- Density-based Method
- Grid-Based Method
- Model-Based Method
- Constraint-based Method

### Partitioning Method

Suppose we are given a database of ‘n’ objects and the partitioning method constructs ‘k’ partition of data. Each partition will represent a cluster and k &le n. It means that it will classify the data into k groups, which satisfy the following requirements &minus

Each group contains at least one object.

Each object must belong to exactly one group.

For a given number of partitions (say k), the partitioning method will create an initial partitioning.

Then it uses the iterative relocation technique to improve the partitioning by moving objects from one group to other.

### Hierarchical Methods

This method creates a hierarchical decomposition of the given set of data objects. We can classify hierarchical methods on the basis of how the hierarchical decomposition is formed. There are two approaches here &minus

### Agglomerative Approach

This approach is also known as the bottom-up approach. In this, we start with each object forming a separate group. It keeps on merging the objects or groups that are close to one another. It keep on doing so until all of the groups are merged into one or until the termination condition holds.

### Divisive Approach

This approach is also known as the top-down approach. In this, we start with all of the objects in the same cluster. In the continuous iteration, a cluster is split up into smaller clusters. It is down until each object in one cluster or the termination condition holds. This method is rigid, i.e., once a merging or splitting is done, it can never be undone.

### Approaches to Improve Quality of Hierarchical Clustering

Here are the two approaches that are used to improve the quality of hierarchical clustering &minus

Perform careful analysis of object linkages at each hierarchical partitioning.

Integrate hierarchical agglomeration by first using a hierarchical agglomerative algorithm to group objects into micro-clusters, and then performing macro-clustering on the micro-clusters.

### Density-based Method

This method is based on the notion of density. The basic idea is to continue growing the given cluster as long as the density in the neighborhood exceeds some threshold, i.e., for each data point within a given cluster, the radius of a given cluster has to contain at least a minimum number of points.

### Grid-based Method

In this, the objects together form a grid. The object space is quantized into finite number of cells that form a grid structure.

The major advantage of this method is fast processing time.

It is dependent only on the number of cells in each dimension in the quantized space.

### Model-based methods

In this method, a model is hypothesized for each cluster to find the best fit of data for a given model. This method locates the clusters by clustering the density function. It reflects spatial distribution of the data points.

This method also provides a way to automatically determine the number of clusters based on standard statistics, taking outlier or noise into account. It therefore yields robust clustering methods.

### Constraint-based Method

In this method, the clustering is performed by the incorporation of user or application-oriented constraints. A constraint refers to the user expectation or the properties of desired clustering results. Constraints provide us with an interactive way of communication with the clustering process. Constraints can be specified by the user or the application requirement.