Home > Articles > Programming > Graphic Programming

  • Print
  • + Share This
This chapter is from the book

Primitive Assembly, Clipping, and Rasterization

After the front end of the pipeline has run (which includes vertex shading, tessellation, and geometry shading), a fixed-function part of the pipeline performs a series of tasks that take the vertex representation of our scene and convert it into a series of pixels, which in turn need to be colored and written to the screen. The first step in this process is primitive assembly, which is the grouping of vertices into lines and triangles. Primitive assembly still occurs for points, but it is trivial in that case.

Once primitives have been constructed from their individual vertices, they are clipped against the displayable region, which usually means the window or screen, but can also be a smaller area known as the viewport. Finally, the parts of the primitive that are determined to be potentially visible are sent to a fixed-function subsystem called the rasterizer. This block determines which pixels are covered by the primitive (point, line, or triangle) and sends the list of pixels on to the next stage—that is, fragment shading.


As vertices exit the front end of the pipeline, their position is said to be in clip space. This is one of the many coordinate systems that can be used to represent positions. You may have noticed that the gl_Position variable that we have written to in our vertex, tessellation, and geometry shaders has a vec4 type, and that the positions we have produced by writing to it are all four-component vectors. This is what is known as a homogeneous coordinate. The homogeneous coordinate system is used in projective geometry because much of the math ends up being simpler in homogeneous coordinate space than it does in regular Cartesian space. Homogeneous coordinates have one more component than their equivalent Cartesian coordinate, which is why our three-dimensional position vector is represented as a four-component variable.

Although the output of the front end is a four-component homogeneous coordinate, clipping occurs in Cartesian space. Thus, to convert from homogeneous coordinates to Cartesian coordinates, OpenGL performs a perspective division, which involves dividing all four components of the position by the last, w component. This has the effect of projecting the vertex from the homogeneous space to the Cartesian space, leaving w as 1.0. In all of the examples so far, we have set the w component of gl_Position as 1.0, so this division has not had any effect. When we explore projective geometry in a short while, we will discuss the effect of setting w to values other than 1.0.

After the projective division, the resulting position is in normalized device space. In OpenGL, the visible region of normalized device space is the volume that extends from −1.0 to 1.0 in the x and y dimensions and from 0.0 to 1.0 in the z dimension. Any geometry that is contained in this region may become visible to the user and anything outside of it should be discarded. The six sides of this volume are formed by planes in three-dimensional space. As a plane divides a coordinate space in two, the volumes on each side of the plane are called half-spaces.

Before passing primitives on to the next stage, OpenGL performs clipping by determining which side of each of these planes the vertices of each primitive lie on. Each plane effectively has an “outside” and an “inside.” If a primitive’s vertices all lie on the “outside” of any one plane, then the whole thing is thrown away. If all of primitive’s vertices are on the “inside” of all the planes (and therefore inside the view volume), then it is passed through unaltered. Primitives that are partially visible (which means that they cross one of the planes) must be handled specially. More details about how this works is given in the “Clipping” section in Chapter 7.

Viewport Transformation

After clipping, all of the vertices of the geometry have coordinates that lie between −1.0 and 1.0 in the x and y dimensions. Along with a z coordinate that lies between 0.0 and 1.0, these are known as normalized device coordinates. However, the window that you’re drawing to has coordinates that usually1 start from (0, 0) at the bottom left and range to (w − 1,h − 1), where w and h are the width and height of the window in pixels, respectively. To place your geometry into the window, OpenGL applies the viewport transform, which applies a scale and offset to the vertices’ normalized device coordinates to move them into window coordinates. The scale and bias to apply are determined by the viewport bounds, which you can set by calling glViewport() and glDepthRange(). Their prototypes are

void glViewport(GLint x, GLint y, GLsizei width, GLsizei height);


void glDepthRange(GLdouble nearVal, GLdouble farVal);

This transform takes the following form:


Here, xw, yw, and zw are the resulting coordinates of the vertex in window space, and xd, yd, and zd are the incoming coordinates of the vertex in normalized device space. px and py are the width and height of the viewport in pixels, and n and f are the near and far plane distances in the z coordinate, respectively. Finally, ox, oy, and oz are the origins of the viewport.


Before a triangle is processed further, it may be optionally passed through a stage called culling, which determines whether the triangle faces toward or away from the viewer and can decide whether to actually go ahead and draw it based on the result of this computation. If the triangle faces toward the viewer, then it is considered to be front-facing; otherwise, it is said to be back-facing. It is very common to discard triangles that are back-facing because when an object is closed, any back-facing triangle will be hidden by another front-facing triangle.

To determine whether a triangle is front- or back-facing, OpenGL will determine its signed area in window space. One way to determine the area of a triangle is to take the cross product of two of its edges. The equation for this is


Here, 042equ02.jpg and 042equ03.jpg are the coordinates of the ith vertex of the triangle in window space and i ⊕ 1 is (i +1) mod 3. If the area is positive, then the triangle is considered to be front-facing; if it is negative, then it is considered to be back-facing. The sense of this computation can be reversed by calling glFrontFace() with dir set to either GL_CW or GL_CCW (where CW and CCW stand for clockwise and counterclockwise, respectively). This is known as the winding order of the triangle, and the clockwise or counterclockwise terms refer to the order in which the vertices appear in window space. By default, this state is set to GL_CCW, indicating that triangles whose vertices are in counterclockwise order are considered to be front-facing and those whose vertices are in clockwise order are considered to be back-facing. If the state is GL_CW, then a is simply negated before being used in the culling process. Figure 3.3 shows this pictorially for the purpose of illustration.

Figure 3.3

Figure 3.3: Clockwise (left) and counterclockwise (right) winding order

Once the direction that the triangle is facing has been determined, OpenGL is capable of discarding either front-facing, back-facing, or even both types of triangles. By default, OpenGL will render all triangles, regardless of which way they face. To turn on culling, call glEnable() with cap set to GL_CULL_FACE. When you enable culling, OpenGL will cull back-facing triangles by default. To change which types of triangles are culled, call glCullFace() with face set to GL_FRONT, GL_BACK, or GL_FRONT_AND_BACK.

As points and lines don’t have any geometric area,2 this facing calculation doesn’t apply to them and they can’t be culled at this stage.


Rasterization is the process of determining which fragments might be covered by a primitive such as a line or a triangle. There are myriad algorithms for doing this, but most OpenGL systems will settle on a half-space–based method for triangles, as it lends itself well to parallel implementation. Essentially, OpenGL will determine a bounding box for the triangle in window coordinates and test every fragment inside it to determine whether it is inside or outside the triangle. To do this, it treats each of the triangle’s three edges as a half-space that divides the window in two.

Fragments that lie on the interior of all three edges are considered to be inside the triangle and fragments that lie on the exterior of any of the three edges are considered to be outside the triangle. Because the algorithm to determine which side of a line a point lies on is relatively simple and is independent of anything besides the position of the line’s endpoints and of the point being tested, many tests can be performed concurrently, providing the opportunity for massive parallelism.

  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

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


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about 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.


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.


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


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


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


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.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. 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.informit.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.


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