This presen

1. View frustum in the context of
head tracking
How to define a correct frustum, thus
the projection matrix?
Naëm Baron - 2014

2. Regular frustum
Far clip distance
Near clip distance
Frustum
Camera/Head
Top View
A frustum illustrate a Projection matrix usually defined by :
near clip distance, far clip distance, FOV and aspect ratio.
Naëm Baron - 2014

3. Projection matrix P
The Projection matrix P is usually defined with the following relations :
Naëm Baron - 2014
𝑷 = 𝑓 𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝐹𝑂𝑉𝑦, 𝑟𝑎𝑡𝑖𝑜
or
𝑷 = 𝑓 𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑛𝑒𝑎𝑟𝑃𝑙𝑎𝑛𝑒𝑊𝑖𝑑𝑡ℎ, 𝑛𝑒𝑎𝑟𝑃𝑙𝑎𝑛𝑒𝐻𝑒𝑖𝑔ℎ𝑡

4. Frustum’s extent
Left
extent
Right
extent
Top View
Far clip distance
Near clip distance
Camera/Head
A Frustum can also be defined using the “extents” at near clip distance in place of the FOV + aspect
ratio.
Naëm Baron - 2014

5. Frustum’s extent
Left
extent
Front View
Right
extent
Bottom
extent
Top
extent
Left/right/top/bottom extent is the distance from the center…
in the left/right/top/bottom direction at the near clip distance.
Far clip plane
Near clip plane
Naëm Baron - 2014

6. Projection matrix P
A new relations to define P:
Naëm Baron - 2014
𝑷 = 𝑓 𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝐹𝑂𝑉𝑦, 𝑟𝑎𝑡𝑖𝑜
or
𝑷 = 𝑓 𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑛𝑒𝑎𝑟𝑃𝑙𝑎𝑛𝑒𝑊𝑖𝑑𝑡ℎ, 𝑛𝑒𝑎𝑟𝑃𝑙𝑎𝑛𝑒𝐻𝑒𝑖𝑔ℎ𝑡
or
𝑷 = 𝑓
𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒,
𝑙𝑒𝑓𝑡𝐸𝑥𝑡𝑒𝑛𝑡, 𝑟𝑖𝑔ℎ𝑡𝐸𝑥𝑡𝑒𝑛𝑡, 𝑡𝑜𝑝𝐸𝑥𝑡𝑒𝑛𝑡, 𝑏𝑜𝑡𝑡𝑜𝑚𝐸𝑥𝑡𝑒𝑛𝑡

7. Adding the screen
Far clip distance
Near clip distance
Camera/Head
Screen
Top View
In immersive virtual reality system like CAVE™ or Wall, the frustum has to “fit” to the screen…
Naëm Baron - 2014

8. Adding the screen
Far clip distance
Top View
In immersive virtual reality system like CAVE™ or Wall, the frustum has to “fit” to the screen…
…wherever the Camera/Head is positioned.
Naëm Baron - 2014
Extents: Left ≠ Right
Asymmetric frustum

9. Projection matrix P
Which function to choose ?
Naëm Baron - 2014
𝑷 = 𝑓 𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝐹𝑂𝑉𝑦, 𝑟𝑎𝑡𝑖𝑜
or
𝑷 = 𝑓 𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑛𝑒𝑎𝑟𝑃𝑙𝑎𝑛𝑒𝑊𝑖𝑑𝑡ℎ, 𝑛𝑒𝑎𝑟𝑃𝑙𝑎𝑛𝑒𝐻𝑒𝑖𝑔ℎ𝑡
or
𝑷 = 𝑓
𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑓𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒,
𝑙𝑒𝑓𝑡𝐸𝑥𝑡𝑒𝑛𝑡, 𝑟𝑖𝑔ℎ𝑡𝐸𝑥𝑡𝑒𝑛𝑡, 𝑡𝑜𝑝𝐸𝑥𝑡𝑒𝑛𝑡, 𝑏𝑜𝑡𝑡𝑜𝑚𝐸𝑥𝑡𝑒𝑛𝑡
Symmetric frustum only
Asymmetric frustum 
How to define the extents according to the head and screen positions ?
How to compute P using the extents ?

10. To get the projection matrix according to the head position :
1. Define the head position relative to the screen center.
 ℎ𝑒𝑎𝑑𝑃𝑜𝑠𝑆𝑐𝑟𝑒𝑒𝑛 = 𝑠𝑐𝑟𝑒𝑒𝑛. 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝑇𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 ℎ𝑒𝑎𝑑𝑃𝑜𝑠𝑊𝑜𝑟𝑙𝑑 *
2. Using the screen size (width, height) you can define the frustum’s extent at the screen distance.
 𝑠𝑐𝑟𝑒𝑒𝑛𝐸𝑥𝑡𝑒𝑛𝑡. 𝐿𝑒𝑓𝑡 = −
𝑤𝑖𝑑𝑡ℎ
2
+ ℎ𝑒𝑎𝑑𝑃𝑜𝑠𝑆𝑐𝑟𝑒𝑒𝑛. 𝑋 **
3. Convert the frustum’s extent at screen distance in frustum’s extent at near clip distance.
 Using Intercept theorem:
𝑒𝑥𝑡𝑒𝑛𝑡. 𝐿𝑒𝑓𝑡 =
𝑛𝑒𝑎𝑟𝐶𝑙𝑖𝑝𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑠𝑐𝑟𝑒𝑒𝑛𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
∗ 𝑠𝑐𝑟𝑒𝑒𝑛𝐸𝑥𝑡𝑒𝑛𝑡. 𝐿𝑒𝑓𝑡
4. Compute the Projection matrix using the near clip extents.
____________________________________________________________________________________________________________
projMtx.m00 = (2.0f * nearClipDistance) / (extent.Right - extent.Left);
projMtx.m02 = (extent.Right + extent.Left) / (extent.Right - extent.Left);
projMtx.m11 = (2.0f * nearClipDistance) / (extent.Top - extent.Bottom);
projMtx.m12 = (extent.Top + extent.Bottom) / (extent.Top - extent.Bottom);
projMtx.m22 = -(farClipDistance + nearClipDistance) / (farClipDistance - nearClipDistance);
projMtx.m23 = (-2.0f * farClipDistance * nearClipDistance) / (farClipDistance - nearClipDistance);
projMtx.m32 = -1.0f;
____________________________________________________________________________________________________________
P from head and screen positions
* InverseTransform convert position from world space to screen local space
** X, Y or Z depending on the coordinate system
Naëm Baron - 2014

11. Intercept Theorem ?
A
B
C
D
E
𝐷𝐸 ‖ 𝐵𝐶

𝐴𝐷
𝐷𝐵
=
𝐴𝐸
𝐸𝐶
and
𝑫𝑬
𝑩𝑪
=
[𝑨𝑫]
[𝑨𝑩]
Math proof: http://math.stackexchange.com/questions/92929/similar-triangle-theorem-in-the-incommensurable-case/92933#92933
Simple explanation:
∠BAC and ∠DAE are “proportionally” the same.
Thus, the sides are also “proportionally” the same.
In fine, the respective quotients of the sides are equals.
Naëm Baron - 2014

12. Back to frustum
Top View
We can use the “Intercept Theorem” to convert the screen extents to near clip extents.
A
BC
DE
𝐴𝐷 = 𝑁𝑒𝑎𝑟 𝑐𝑙𝑖𝑝 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝐴𝐵 = 𝑆𝑐𝑟𝑒𝑒𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝐵𝐶 = 𝐿𝑒𝑓𝑡 𝑠𝑐𝑟𝑒𝑒𝑛′ 𝑠 𝑒𝑥𝑡𝑒𝑛𝑡
𝑫𝑬 = ?
𝐷𝐸 =
𝐴𝐷
𝐴𝐵
∗ 𝐵𝐶
𝐷𝐸 is the left extent at near clip distance.
Naëm Baron - 2014

13. Final note
At each tracking frame (not rendering frame):
Naëm Baron - 2014
1
• Define the head position relative to the
screen center.
2
• Define the frustum’s extent at the screen
distance.
3
• Extent at screen distance  Extent at near
clip distance.
4
• Compute Projection matrix
Left, Right, Top, Bottom
Left, Right, Top, Bottom

14. return;
Naëm Baron
Naem.Baron—gmail•com
Additional resources:
http://paulbourke.net/stereographics/stereorender/
http://schabby.de/projection-matrix/
http://simply3d.wordpress.com/2009/05/30/perspective-projection-part-3/
http://msdn.microsoft.com/en-us/library/windows/desktop/bb205353(v=vs.85).aspx
http://msdn.microsoft.com/en-us/library/windows/desktop/dd373537(v=vs.85).aspx