Publishers of technology books, eBooks, and videos for creative people

Home > Articles

Geometry for Modeling and Design

This chapter is from the book

Many different geometric shapes were used to model this jetboard. The wireframe view of the top cover reveals several regular geometric shapes used to model the interior components. The graceful lines of the outer hull are defined by the irregular curves used to model it. (Courtesy of Leo Greene, www.e-Cognition.net.)

Coordinates for 3D CAD Modeling

2D and 3D CAD drawing entities are stored in relationship to a Cartesian coordinate system. No matter what CAD software system you will be using, it is helpful to understand some basic similarities of coordinate systems.

Most CAD systems use the right-hand rule for coordinate systems; if you point the thumb of your right hand in the positive direction for the X-axis and your index finger in the positive direction for the Y-axis, your remaining fingers will curl in the positive direction for the Z-axis (shown in Figure 4.1). When the face of your monitor is the X-Y plane, the Z-axis is pointing toward you (see Figure 4.2).

4.1

4.1 Right-Hand Rule

4.2

4.2 The Z-Axis. In systems that use the right-hand rule, the positive Z-axis points toward you when the face of the monitor is parallel to the X-Y plane.

The right-hand rule is also used to determine the direction of rotation. For rotation using the right-hand rule, point your thumb in the positive direction along the axis of rotation. Your fingers will curl in the positive direction for the rotation, as shown in Figure 4.3.

4.3

4.3 Axis of Rotation. The curl of the fingers indicates the positive direction along the axis of rotation.

Though rare, some CAD systems use a left-hand rule. In this case, the curl of the fingers on your left hand gives you the positive direction for the Z-axis. In this case, when the face of your computer monitor is the X-Y plane, the positive direction for the Z-axis extends into your computer monitor, not toward you.

A 2D CAD system uses only the X- and Y-coordinates of the Cartesian coordinate system. 3D CAD systems use X, Y, and Z. To represent 2D in a 3D CAD system, the view is straight down the Z-axis. Figure 4.4 shows a drawing created using only the X- and Y- values, leaving the Z-coordinates set to 0, to produce a 2D drawing.

4.4

4.4 2D CAD Drawing. This drawing was created on the X-Y plane in the CAD system. It appears true shape because the viewing direction is perpendicular to the X-Y plane—straight down the Z-axis.

Recall that each orthographic view shows only two of the three coordinate directions because the view is straight down one axis. 2D CAD drawings are the same: They show only the X- and Y-coordinates because you are looking straight down the Z-axis.

When the X-Y plane is aligned with the screen in a CAD system, the Z-axis is oriented horizontally. In machining and many other applications, the Z-axis is considered to be the vertical axis. In all cases, the coordinate axes are mutually perpendicular and oriented according to the right-hand or left-hand rule. Because the view can be rotated to be straight down any axis or any other direction, understanding how to use coordinates in the model is more important than visualizing the direction of the default axes and planes.

The vertices of the 3D shape shown in Figure 4.5 are identified by their X-, Y-, and Z-coordinates. Often, it is useful when modeling parts to locate the origin of the coordinate system at the lower left of the part, as shown in Figure 4.5. This location for the (0,0,0) point on a part is useful when the part is being machined, as it then makes all coordinates on the part positive (Figure 4.6). Some older numerically-controlled machinery will not interpret a file correctly if it has negative lengths or coordinates. CAD models are often exported to other systems for manufacturing parts, so try to create them in a common and useful way.

4.5

4.5 3D Coordinates for Vertices

4.6

4.6 This CAD model for a plate with 6 holes has its origin (0,0,0) at the back left of the part when it is set up for numerically-controlled machining. (Courtesy of Matt McCune, Autopilot, Inc.)

4.7

4.7 The part is clamped in place during machining. The back left corner of the part is the 0,0,0 location during the machining process. (Courtesy of Matt McCune, Autopilot, Inc.)

Specifying Location

Even though the model is ultimately stored in a single Cartesian coordinate system, you may usually specify the location of features using other location methods as well. The most typical of these are relative, polar, cylindrical, and spherical coordinates. These coordinate formats are useful for specifying locations to define your CAD drawing geometry.

Absolute Coordinates

Absolute coordinates are used to store the locations of points in a CAD database. These coordinates specify location in terms of distance from the origin in each of the three axis directions of the Cartesian coordinate system.

Think of giving someone directions to your house (or to a house in an area where the streets are laid out in rectangular blocks). One way to describe how to get to your house would be to tell the person how many blocks over and how many blocks up it is from two main streets (and how many floors up in the building, for 3D). The two main streets are like the X- and Y-axes of the Cartesian coordinate system, with the intersection as the origin. Figure 4.8 shows how you might locate a house with this type of absolute coordinate system.

4.8

4.8 Absolute coordinates define a location in terms of distance from the origin (0,0,0), shown here as a star. These directions are useful because they do not change unless the origin changes.

Relative Coordinates

Instead of having to specify each location from the origin, you can use relative coordinates to specify a location by giving the number of units from a previous location. In other words, the location is defined relative to your previous location.

To understand relative coordinates, think about giving someone directions from his or her current position, not from two main streets. Figure 4.9 shows the same map again, but this time with the location of the house relative to the location of the person receiving directions.

4.9

4.9 Relative coordinates describe the location in terms of distance from a starting point. Relative coordinates to the same location differ according to the starting location.

Polar Coordinates

Polar coordinates are used to locate an object by giving an angle (from the X-axis) and a distance. Polar coordinates can either be absolute, giving the angle and distance from the origin, or relative, giving the angle and distance from the current location.

Picture the same situation of having to give directions. You could tell the person to walk at a specified angle from the crossing of the two main streets, and how far to walk. Figure 4.10 shows the angle and direction for the shortcut across the empty lot using absolute polar coordinates. You could also give directions as an angle and distance relative to a starting point.

4.10

4.10 Polar coordinates describe the location using an angle and distance from the origin (absolute) or starting point (relative).

Cylindrical and Spherical Coordinates

Cylindrical and spherical coordinates are similar to polar coordinates except that a 3D location is specified instead of one on a single flat plane (such as a map).

Cylindrical coordinates specify a 3D location based on a radius, angle, and distance (usually in the Z-axis direction). This gives a location as though it were on the edge of a cylinder. The radius tells how far the point is from the center (or origin); the angle is the angle from the X-axis along which the point is located; and the distance provides the height where the point is located on the cylinder. Cylindrical coordinates are similar to polar coordinates, but they add distance in the Z-direction. Figure 4.11a depicts relative cylindrical coordinates used to specify a location, where the starting point serves as the center of the cylinder.

4.11

4.11 Relative Cylindrical and Spherical Coordinates. The target points in (a) and (b) are described by relative coordinates from the starting point (3,2,0). Although the paths to the point differ, the resulting endpoint is the same.

Spherical coordinates specify a 3D location by the radius, an angle from the X-axis, and the angle from the X-Y plane. These coordinates locate a point on a sphere, where the origin of the coordinate system is at the center of the sphere. The radius gives the size of the sphere; the angle from the X-axis locates a place on the equator. The second angle gives the location from the plane of the equator to a point on the sphere in line with the location specified on the equator. Figure 4.11b depicts relative spherical coordinates, where the starting point serves as the center of the sphere.

Even though you may use these different systems to enter information into your 3D drawings, the end result is stored using one set of Cartesian coordinates.

Using Existing Geometry to Specify Location

Most CAD packages offer a means of specifying location by specifying a the relationship of a point to existing objects in the model or drawing. For example, AutoCAD’s “object snap” feature lets you enter a location by “snapping” to the endpoint of a line, the center of a circle, the intersection of two lines, and so on (Figure 4.12). Using existing geometry to locate new entities is faster than entering coordinates. This feature also allows you to capture geometric relationships between objects without calculating the exact location of a point. For example, you can snap to the midpoint of a line or the nearest point of tangency on a circle. The software calculates the exact location.

4.12

4.12 Object snaps are aids for selecting locations on existing CAD drawing geometry. (Autodesk screen shots reprinted courtesy of Autodesk, Inc.)

Peachpit Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from Peachpit and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about Peachpit products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites; develop new products and services; conduct educational research; and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email ask@peachpit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by Adobe Press. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.peachpit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020