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To prevent artifacts when geometry penetrates a physical boundary plane, an extra clipping plane is typically needed beyond the standard six-sided view frustum. This paper proposes modifying the projection matrix so the conventional near plane is repositioned to act as an oblique clipping boundary, keeping the total clipping planes at six. It analyzes the impact on the far plane and depth buffer precision and derives an optimal oblique view frustum construction.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/oblique-view-frustum-depth-projection-and-clipping/648575/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/oblique-view-frustum-depth-projection-and-clipping/648575.png","ImageObject",300,407,{"name":92,"@type":93},"asda1da","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-25","2026-10-23",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does the paper address in recursive 3D rendering?","Question",{"text":112,"@type":113},"Recursive rendering often requires a virtual camera and a boundary plane, but geometry can cross that boundary and cause unwanted artifacts in the final image.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the proposed method avoid using an extra clipping plane?",{"text":117,"@type":113},"It modifies the projection matrix so the conventional near plane is repositioned to coincide with the oblique clipping boundary, keeping the total clipping planes at six.",{"name":119,"@type":110,"acceptedAnswer":120},"What are the consequences of moving the near plane?",{"text":121,"@type":113},"The four side planes remain unaffected, but the far plane is destroyed and depth buffer precision is adversely impacted, so the paper analyzes these effects and provides a construction method for an optimal oblique frustum.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},648575,1792898414,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},3985747861852,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Oblique View Frustum  \nDepth Projection and Clipping  \nEric Lengyel  \nTerathon Software  \n[lengyel@terathon.com](lengyel@terathon.com)  \nAbstract. Several 3D rendering techniques have been developed in which part of the final image is the result of rendering from a virtual camera whose position in the scene differs from that of the primary camera. In these situations, there is usually a planar surface, such as the reflecting plane of a mirror, that can be considered the physical boundary of the recursively rendered image. In order to avoid artifacts that can arise when rendered geometry penetrates the boundary plane from the perspective of the virtual camera, an additional clipping plane must be added to the standard six-sided view frustum. However, many 3D graphics processors cannot support an extra clipping plane natively, or require that vertex and fragment shaders be augmented to explicitly perform the additional clipping operation.  \nThis paper discusses a technique that modifies the projection matrix in such a way that the conventional near plane of the view frustum is repositioned to serve as the generally oblique boundary clipping plane. Doing so avoids the performance penalty and burden of developing multiple shaders associated with user-defined clipping planes by keeping the total number of clipping planes at six. The near plane is moved without affecting the four side planes, but the conventional far plane is inescapably destroyed. We analyze the effect on the far plane as well as the related impact on depth buffer precision and present a method for constructing the optimal oblique view frustum.  \n1. Introduction  \nSeveral techniques have been developed to render 3D images containing elements that are inherently recursive in nature. Some examples are mirrors that reflect their immediate surroundings, portals through which a remote region of the scene can be viewed, and water surfaces through which refractive transparency is applied. Each of these situations requires that part of the scene be rendered from the perspective of some virtual camera whose position and orientation are calculated using certain rules that take into account the position of the primary camera through which the user is looking. For example, the image visible in a mirror is rendered using a camera that is the reflection of the primary camera through the plane of the mirror.  \nPublished in Journal of Game Development, Vol. 1, No. 2 (March 2005), pp. 5–16. 1  \n2  \nOnce such a component of an image is rendered through a virtual camera, it is usually treated as a geometrically planar object when rendering from the perspective of the primary camera. The plane chosen to represent the image is simply the plane that naturally separates the image from the rest of the environment, such as the plane of a mirror, portal, or water surface. In the process of rendering from a virtual camera, it is possible that geometry lies closer to the camera than the plane representing the surface of the mirror, portal, etc. If such geometry is rendered, it can lead to unwanted artifacts in the final image.  \nThe simplest solution to this problem is to enable a user-defined clipping plane to truncate all geometry at the surface. Unfortunately, older GPUs do not support userdefined clipping planes in hardware and must resort to a software-based vertex processing path when they are enabled. Later GPUs do support generalized user-defined clipping operations, but using them requires that the vertex or fragment programs be modified—a task that may not be convenient since it necessitates two versions of each program be kept around to render a particular geometry.  \nThis article presents an alternative solution that exploits the clipping planes that already exist for every rendered scene. Normally, every geometric primitive is clipped to the six sides of the view frustum: four side planes, a near plane, and a far plane. Adding a seventh clipping plane that represents the surface ","cbCaitbyeOI5GQQv","https://ap.wps.com/l/cbCaitbyeOI5GQQv","pdf",583829,11,"English","# Introduction\n# Background","[{\"question\":\"What problem does the paper address in recursive 3D rendering?\",\"answer\":\"Recursive rendering often requires a virtual camera and a boundary plane, but geometry can cross that boundary and cause unwanted artifacts in the final image.\"},{\"question\":\"How does the proposed method avoid using an extra clipping plane?\",\"answer\":\"It modifies the projection matrix so the conventional near plane is repositioned to coincide with the oblique clipping boundary, keeping the total clipping planes at six.\"},{\"question\":\"What are the consequences of moving the near plane?\",\"answer\":\"The four side planes remain unaffected, but the far plane is destroyed and depth buffer precision is adversely impacted, so the paper analyzes these effects and provides a construction method for an optimal oblique frustum.\"}]","Oblique View Frustum - Depth Projection and Clipping | PDF",1792794075,28]