Pictures in Excel have way too much power…

You’d think inserting a picture into an Excel worksheet is pretty straightforward. If you’re just inserting the picture, then it is.

Then you check what other properties you can play with manipulating the picture. Woah amigo! There’s a whole plethora of options!

Excel picture format options

That’s like giving the typical user a (good enough) subset of rendering functions, including 3D. And here I thought you could just resize the picture, and maybe rotate it… If you’re not going for full 3D manipulation as in OpenGL or DirectX, Excel will do it for you.

Excel respects alpha/transparency of the picture (right now, I only know of PNG images that contain alpha information AND is fairly prevalent). And you can outline the picture and fill the transparent portions with a colour.

SpreadsheetLight picture fill and outline

You can even do shadows!

SpreadsheetLight picture shadows

Do you know it can take a phenomenal amount of work to get shadows working in 3D renderings? You’ve got to know the light source position, calculate the rays of light blocked by an object to create the illusion of a shadow (by creating a polygon that’s the outline of that object as described by those blocked rays of light). Well, technically speaking, Excel has got it easy, since the object is always a rectangle (an image).

Speaking of light, we can also do reflections. Let’s run a full gamut of the 3D power.

SpreadsheetLight 3D pictures

I lost the software keys to my 2 3D rendering software (Bryce 4 and TrueSpace). It sucked that I couldn’t render landscapes anymore… But, I’m seeing Excel as an alternative to getting some 3D-ish images.

Well, the above screenshots were from a spreadsheet generated by my software library, SpreadsheetLight. The library is based on the Open XML SDK, which in turn is based on what Excel can do. Here’s how the source code looks like:

SLDocument sl = new SLDocument(SLThemeTypeValues.Oriel);

SLPicture pic;
                
sl.SetCellValue(2, 2, "Accent 1");
sl.ApplyNamedCellStyle(2, 2, SLNamedCellStyleValues.Accent1);
sl.SetCellValue(2, 3, "Accent 2");
sl.ApplyNamedCellStyle(2, 3, SLNamedCellStyleValues.Accent2);
sl.SetCellValue(2, 4, "Accent 3");
sl.ApplyNamedCellStyle(2, 4, SLNamedCellStyleValues.Accent3);
sl.SetCellValue(2, 5, "Accent 4");
sl.ApplyNamedCellStyle(2, 5, SLNamedCellStyleValues.Accent4);
sl.SetCellValue(2, 6, "Accent 5");
sl.ApplyNamedCellStyle(2, 6, SLNamedCellStyleValues.Accent5);
sl.SetCellValue(2, 7, "Accent 6");
sl.ApplyNamedCellStyle(2, 7, SLNamedCellStyleValues.Accent6);

pic = new SLPicture("mandelbrot.png");
// anchor at cell (4,2) with 0 horizontal and vertical offsets.
pic.SetRelativePositionInPixels(4, 2, 0, 0);
sl.InsertPicture(pic);

pic = new SLPicture("mandelbrot.png");
pic.SetRelativePositionInPixels(4, 6, 0, 0);
pic.SetSolidFill(DocumentFormat.OpenXml.Drawing.SchemeColorValues.Accent3, 20);
pic.SetSolidOutline(DocumentFormat.OpenXml.Drawing.SchemeColorValues.Accent4, 0);
pic.SetOutlineStyle(4, DocumentFormat.OpenXml.Drawing.CompoundLineValues.Double, DocumentFormat.OpenXml.Drawing.PresetLineDashValues.LargeDash, DocumentFormat.OpenXml.Drawing.LineCapValues.Round, SLPicture.SLPictureJoinType.Bevel);
sl.InsertPicture(pic);

sl.AddWorksheet("Sheet2");

sl.SetCellValue(2, 2, "Accent 1");
sl.ApplyNamedCellStyle(2, 2, SLNamedCellStyleValues.Accent1);
sl.SetCellValue(2, 3, "Accent 2");
sl.ApplyNamedCellStyle(2, 3, SLNamedCellStyleValues.Accent2);
sl.SetCellValue(2, 4, "Accent 3");
sl.ApplyNamedCellStyle(2, 4, SLNamedCellStyleValues.Accent3);
sl.SetCellValue(2, 5, "Accent 4");
sl.ApplyNamedCellStyle(2, 5, SLNamedCellStyleValues.Accent4);
sl.SetCellValue(2, 6, "Accent 5");
sl.ApplyNamedCellStyle(2, 6, SLNamedCellStyleValues.Accent5);
sl.SetCellValue(2, 7, "Accent 6");
sl.ApplyNamedCellStyle(2, 7, SLNamedCellStyleValues.Accent6);

pic = new SLPicture("mandelbrot.png");
pic.SetRelativePositionInPixels(4, 2, 0, 0);
// transparency 0%, blur 6pt, angle 90 degrees, distance 3pt
pic.SetInnerShadow(DocumentFormat.OpenXml.Drawing.SchemeColorValues.Accent1, 0, 6, 90, 3);
sl.InsertPicture(pic);

pic = new SLPicture("mandelbrot.png");
pic.SetRelativePositionInPixels(4, 6, 0, 0);
// transparency 50%, horizontal size 100%, vertical size -23%, horizontal skew -13.34%, vertical skew 0%
// blur 6pt, angle 45 degrees, distance 3pt
// origin at picture's bottom left, don't rotate with picture.
pic.SetPerspectiveShadow(DocumentFormat.OpenXml.Drawing.SchemeColorValues.Accent2, 50, 100, -23, -13.34m, 0, 6, 45, 3, DocumentFormat.OpenXml.Drawing.RectangleAlignmentValues.BottomLeft, false);
sl.InsertPicture(pic);

sl.AddWorksheet("Sheet3");

sl.SetCellValue(2, 2, "Accent 1");
sl.ApplyNamedCellStyle(2, 2, SLNamedCellStyleValues.Accent1);
sl.SetCellValue(2, 3, "Accent 2");
sl.ApplyNamedCellStyle(2, 3, SLNamedCellStyleValues.Accent2);
sl.SetCellValue(2, 4, "Accent 3");
sl.ApplyNamedCellStyle(2, 4, SLNamedCellStyleValues.Accent3);
sl.SetCellValue(2, 5, "Accent 4");
sl.ApplyNamedCellStyle(2, 5, SLNamedCellStyleValues.Accent4);
sl.SetCellValue(2, 6, "Accent 5");
sl.ApplyNamedCellStyle(2, 6, SLNamedCellStyleValues.Accent5);
sl.SetCellValue(2, 7, "Accent 6");
sl.ApplyNamedCellStyle(2, 7, SLNamedCellStyleValues.Accent6);

pic = new SLPicture("mandelbrot.png");
pic.SetRelativePositionInPixels(6, 3, 0, 0);
pic.SetFullReflection();
// width 6pt, height 6pt
pic.Set3DBevelBottom(DocumentFormat.OpenXml.Drawing.BevelPresetValues.Convex, 6, 6);
// width 3pt, height 4pt
pic.Set3DBevelTop(DocumentFormat.OpenXml.Drawing.BevelPresetValues.ArtDeco, 3, 4);
// extrusion colour transparency 0%, extrusion (or depth) height 15 pt
pic.Set3DExtrusion(System.Drawing.Color.Green, 0, 15);
// contour colour transparency 40%, contour width 4pt
pic.Set3DContour(DocumentFormat.OpenXml.Drawing.SchemeColorValues.Accent3, 40, 4);
pic.Set3DMaterialType(DocumentFormat.OpenXml.Drawing.PresetMaterialTypeValues.TranslucentPowder);
// 5 pt above "ground"
pic.Set3DZDistance(5);
// field of view 105 degrees, zoom 100%
// camera latitude, longitude, revolution in degrees (50, 40, 30)
// light rig latitude, longitude, revolution in degrees (0, 0, 30)
pic.Set3DScene(DocumentFormat.OpenXml.Drawing.PresetCameraValues.PerspectiveFront, 105, 100, 50, 40, 30, DocumentFormat.OpenXml.Drawing.LightRigValues.Sunrise, DocumentFormat.OpenXml.Drawing.LightRigDirectionValues.TopLeft, 0, 0, 30);
sl.InsertPicture(pic);

sl.SaveAs("Pictures.xlsx");

I use the enumerations available in Open XML SDK as far as possible. This means you can sort of port over any programs you have written using Open XML SDK. And also that I don’t have to invent my own enumerations that are essentially copies of the SDK enumerations.

One thing that confused me was the 3D rotation options in Excel.

SpreadsheetLight 3D picture rotation options

Excel used the vertical “axis” as the X value, and the horizontal “axis” as the Y value. To the user, the X, Y and Z values are like the first, second and third values of … something.

To me, X, Y and Z have special meanings, particularly when applied to the context of 3D. Those values don’t even increase in a consistent manner. The X value decreases if you use the right-hand rule (the left button), but the Y and Z values increase when you use the right-hand rule (left and right buttons respectively).

The Open XML SDK then complicates matters by using the terms latitude, longitude and revolution. Each ranging from 0 degrees to 359.9 degrees (or strictly less than 360 degrees in other words).

And in case you’re wondering about the state of the spreadsheet library progress, I froze the feature set of SpreadsheetLight for version 1. Otherwise, I’d never get it shipped. I’m now documenting the hashbrown out of the source code. The point is to make it ridiculously easy for a programmer to pick up SpreadsheetLight and start using it (without having to refer to a manual, which I’ll also be writing).

3D Bézier curve editor

Timo Suoranta created a 3D Bézier curve editor. As of this writing, the program runs on Windows and requires OpenGL version 3 or later (shaders are involved). Here’s a screenshot:

Bezier curve editor
[click for larger image]

It looks awesome. What, no? Then you have to understand picking. In 2D, any point you click on the screen is exact. The point you click on is the point selected.

In 3D, it’s different. There are an infinite number of planes behind that virtual screen you picked on. Think of looking up at the clouds in the sky. You know the water droplets are scattered sparsely and densely in the sky. You know they are in a 3D space. But you, looking up at the sky, only see one plane, the 2D plane that has the water droplets rendered onto.

In this case, it’s simpler. We are only concerned with the points on the Bézier curve itself. Timo used the closest point to the clicked point on the screen as the chosen point. Basically you “shoot” a ray from the clicked point into the vanishing point in the far distance (far far far distance, as in infinity far). When your ray hits the Bézier curve, that’s the chosen point. You can find out more about the method by searching on the Internet for “3d picking” or something similar.

So how do you edit a point once it’s chosen? Timo solved it by using 3 cones to represent the X, Y and Z axes. Dragging on the cones move the point along the respective direction of the axis. Notice the 3 cones at the right side.

Bezier curve editor with axes
[click for larger image]

I believe most 3D rendering software use something similar to edit points.

Now notice the small details Timo added:

  • The floor is a checker board to illustrate the notion of space
  • The vertical lines drawn from the points on the curve to the checker board to show the spatial relation
  • If you hover over the big blue spheres, the checker board, or the curve itself, they glow pulse-like

Wait, you haven’t downloaded the program? Here’s the link.

Help! Getting a “nice” reverse engineered Bézier curve

Commenter Timo wants to know how to get a nice shape for a reverse engineered Bézier curve.

The question started from calculating the control points of a cubic Bézier curve if you’re given 4 points that lie on the curve, assuming the first and last given points were also the first and last control points. I wrote a similar article for a quadratic Bézier curve too.

To sum up, you have 1 degree of freedom when working on a quadratic Bézier curve. In the case of a cubic Bézier curve, you have 2 degrees of freedom, meaning 2 variables in the calculation that you have to decide on. Now this posed a problem, because there are an infinite number of solutions. How do you decide on a numeric value?

Well, for a quadratic Bézier curve, the simplest and obvious option is to choose the second given point (since the first and last control points are determinable) to be at the halfway mark. For a cubic Bézier curve, the second and third given points are chosen to be at the 1/3 and 2/3 mark along the curve respectively. Now this may or may not be suitable, but it does give you something to start with.

I want to state right now, that I had not been doing formal mathematics for a while. There is a limit to what I know, and I’m not an expert. I just know enough to figure out how to solve problems. Sometimes, it’s not enough. Keep that in mind.

“Nice curve” is subjective

Now Timo’s problem is getting a better shaped cubic Bézier curve from those calculations. Since the 4 given points are fixed, and the first and last control points are also fixed, the only thing you can manipulate are the second and third control points. Which in turn means deciding on values of u and v to get a “nice” cubic Bézier curve in the end.

This, “niceness”, is a subjective criteria. How do you determine if a cubic Bézier curve looks nice? Remember that we don’t have the control points yet. So we don’t know how the curve looks like. So we don’t even know if manipulating the second and third control point to not be at 1/3 and 2/3 will result in a nice curve. It’s a chicken and egg problem.

Apportioned chord length

During the correspondence with Timo, some solutions were discussed. The next simplest set of values to try for u and v are calculated based on the given points.

Let d1 be the distance between the first and second given point.
Let d2 be the distance between the second and third given point.
Let d3 be the distance between the third and last (fourth) given point.

Then let u = d1/(d1 + d2 + d3) and v = (d1 + d2)/(d1 + d2 + d3). This should result in a curve that’s “better shaped” than the (u,v) pair (1/3, 2/3). When I wrote that article for the cubic curve version, this was the next default set of values for u and v, but I didn’t want to add too much more. Well, nobody asked, so I left it as it was.

It turns out that Don Lancaster already wrote about it. He called it “apportioned chords” method.
http://www.tinaja.com/glib/nubz4pts1.pdf

Inflection points

Then Timo had another problem. He wanted the second and third given points to also lie at the “loop tips”. What are loop tips? After some clarification, I believe Timo is referring to the inflection points on the curve. An inflection point

is a point on a curve at which the curvature (second derivative) changes signs. The curve changes from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa.

A cubic Bézier curve can have 0, 1 or 2 inflection points. If it’s a straight line, it has 0. If it’s U-shaped, it has 1. And if it zigzags, it has 2. And Adrian Colomitchi proved that there are at most 2 inflection points on a cubic Bézier curve.

UPDATE: The following diagram is wrong. Please refer to this article for the correct version. I was thinking of points where the second derivative was zero, not when it changed signs (as defined above).

Inflection points

As you can see, inflection points don’t necessarily have to coincide with the given points.

By the way, I used Paint.NET for the illustration. I took a screenshot of me drawing the curve, still with the given points visible (noted by the small squares). Paint.NET appears to have succeeded in doing the very thing Timo wants, to render a cubic Bézier curve using 4 given points. Of course, I’m assuming the image editor is using cubic Bézier curves…

The math paper

I found another reference with a more explicit mathematical formulation to help Timo.
http://www.cis.usouthal.edu/~hain/general/Publications/Bezier/bezier%20cccg04%20paper.pdf

As of this writing, page 3 of that paper explicitly shows the calculation needed to find the inflection points of a cubic Bézier curve, if they exist. Let me emphasise that again. The curve might have 2 inflection points, only 1 inflection point, or none at all (a straight line, the trivial version of a Bezier curve, has no inflection points).

Maximum curvature

Then Timo found another forum posting for the fast calculation of maximum curvature. An inflection point would have the “maximum curvature”. The problem with that solution is that it assumes we have the control points.

So my suggestion was to do iteration. Use my method to get a set of control points. Perhaps use the apportioned chords to start off with a good set of control points. Then apply the maximum curvature solution to find the inflection points and the associated values of t. With those values of t, pump them back into my method to find a new set of control points. Pump those control points into the maximum curvature solution to find inflection points and their t values. Iterate till the t values between iterations are within acceptable margins of error.

Caveat: I don’t know if this combination of 2 algorithms in an iterative manner will converge. I have not tested this. Use at your own risk.

Scaling up to 3 dimensions

Anyway, Timo found another solution himself (he didn’t say what though). He still needed to handle that cusp point. What cusp point, you ask? It’s in page 3 of that paper I mentioned above. That paper is for 2 dimensional cubic Bézier curves. The degree of the curve is independent of the degree of the dimensions. Timo wants to know how the 3 dimensional case will look like.

Now, the method of finding inflection points is to do a cross product of the first derivative and second derivative of the cubic Bézier curve equation. The Bézier curve is parametrised into

x(t) = ax * t^3 + bx * t^2 + cx * t + dx
y(t) = ay * t^3 + by * t^2 + cy * t + dy

and using the Bézier basis matrix, the coefficients are

ax = -x0 + 3*x1 – 3*x2 + x3
bx = 3*x0 – 6*x1 + 3*x2
cx = -3*x0 + 3*x1
dx = x0

ay = -y0 + 3*y1 – 3*y2 + y3
by = 3*y0 – 6*y1 + 3*y2
cy = -3*y0 + 3*y1
dy = y0

Set the cross product to 0. The inflection points are found at values of t when you solve that equation:

x’ * y” – x” * y’ = 0

where x’ and x” are the first and second derivatives of x(t). Similarly for y’ and y”. The solution is in that paper I mentioned before.

This is if the curve is in 2D. The cross product of 2D vectors is a scalar. And we set the scalar to 0 to solve for t. The cross product of 3D vectors is a vector, and so we’re solving with a zero vector.

So with
z(t) = az * t^3 + bz * t^2 + cz * t + dz
for the third dimension, we have

x’ = 3 * ax * t^2 + 2 * bx * t + cx
y’ = 3 * ay * t^2 + 2 * by * t + cy
z’ = 3 * az * t^2 + 2 * bz * t + cz

x” = 6 * ax * t + 2 * bx
y” = 6 * ay * t + 2 * by
z” = 6 * az * t + 2 * bz

The cross product is the determinant of the following
Cross product

where i, j, k are the unit vectors. I’ll leave it to you to find out the formula for the determinant of a 3 by 3 matrix.

So we’re going to solve this:
0 = (y’ * z’’ – z’ * y’’) * i – (x’ * z’’ – z’ * x’’) * j + (x’ * y’’ – y’ * x’’) * k
where 0 is the zero vector.

This implies that

y’ * z’’ – z’ * y’’ = 0
x’ * z’’ – z’ * x’’ = 0
x’ * y’’ – y’ * x’’ = 0

This time the zeroes are scalars. We now have 3 times the number of equations to solve when compared to the 2D case. This means there are potentially 6 values of t for the inflection points to check. Hopefully, there will be repeated values of t. Hopefully, the number of unique values of t is 2 or less (remember Adrian’s proof?).

If a t value is repeated, it’s probably an inflection point. What if we get 6 unique values? Is a 6-unique-value case even possible? I don’t know. You’ll have to interpret the values in the best way you can, based on some assumptions and arguments.

What do you think?

So after reading through that entire article, what do you think? Comments on the methods I described? Do you have a new method? Your thoughts on whether this problem is even solvable? That I’m a complete idiot?

Let me know in a comment. Or better, write a blog post and tell me about it. Because if it took me that long to explain the solutions, your solution is probably just as long. It’s a Bézier curve. A picture might be appropriate.

Stationary camera, moving scene

Previously, we talked about revolving the entire 3D scene about the camera, and also the problem of the camera looking directly downwards. Today, we’ll look at the mechanics of implementing that stationary camera (it ain’t pretty).

There are 2 transformations to take care of: translation and rotation. Translation takes care of the distance between the camera and the point it’s looking at. Rotation takes care of simulating the camera turning around to look at objects, roughly speaking. Let me use a 2D version to illustrate the concept.

Reverse translation and rotation of 2D scene

Suppose the camera is at some arbitrary position looking at an object. Based on the positions of the camera and the object, you can find the distance between them. You know, with this:
d = sqrt( (cx-ox)^2 + (cy-oy)^2 )
where cx and cy are the x-coordinate and y-coordinate of the camera respectively, and ox and oy are the x-coordinate and y-coordinate of the object respectively.

The camera is looking at the object, so the angle (theta) of its line of sight with respect to the (for example) x-axis can be calculated.

Suppose we want the stationary camera to look in the direction of the positive y-axis, and be positioned at the origin (0,0). To make the scene viewed through a stationary camera the same as that in the original version (the default by the 3D engine), we would rotate the entire scene (90 – theta) degrees, then translate the result of that d units along the positive y-axis.

Remember that order of transformations is important. Rotating first then translating, is (generally) different from translating then rotating.

So that’s the general idea of making a stationary camera work, by moving and rotating the entire scene. The fun part comes because it’s in 3D.

The distance calculation still holds true:
d = sqrt(x^2 + y^2 + z^2)

The angle… not so much. Because it’s in 3D, I adopted spherical coordinates. The radius would simply be the distance calculated previously. But there are now 2 angles to calculate, theta and phi.

Spherical coordinate angles

Suppose the camera is at (a,b,c) and the viewed object is at (p,q,r). We make the viewed object the centre of our attention, so we start our calculations with the object at the origin. Therefore, the camera is at (a-p, b-q, c-r).

We can calculate the distance between them as
d = sqrt( (a-p)^2 + (b-q)^2 + (c-r)^2 )

Then we also solve for the following set of simultaneous equations (note I’m using y-axis as the “upward” axis)
x = r * sin(theta) * sin(phi)
y = r * cos(phi)
z = r * cos(theta) * sin(phi)

==>

a-p = d * sin(theta) * sin(phi)
b-q = d * cos(phi)
c-r = d * cos(theta) * sin(phi)

to look for the angles theta and phi, where
0 <= theta <= 2*PI 0 <= phi < PI Once found, the rendering occurs by rotating the entire scene phi degrees about the positive z-axis (starting from negative y-axis as 0 degrees), then rotate about the positive y-axis (starting from the positive z-axis as 0 degrees), then translate by (-a,-b,-c) (this moves the entire scene away from the camera positioned at the origin). Well, that was a lot of trouble. What was I trying to solve again? Oh yeah, that looking down and losing the "up" vector problem. Notice anything wrong in this implementation? The "up" vector of the camera was never considered. But figuring out all the math was fun... if only it solved something too... *sigh* [Note: all use of "degrees" in this article can be substituted with "radians", depending on your situation. Use accordingly.]

The problem of looking down

So in film, the camera usually moves a lot, together with the actors and props. The scene doesn’t move. In the virtual 3D world, we can move and revolve the world around the camera, which I talked about previously.

Let’s talk about the virtual camera first before launching into the problem I was trying to solve. There are 3 positional vectors for a virtual camera: its position, its “up” vector, and where it’s looking at. The 1st and 3rd positional vector should be easily understood. The “up” vector refers to where the “head” of the camera is pointing at.

You’re looking at something in front of you. Now tilt your head left. Your eyes are still at the same position (1st vector), and you’re still looking at the same thing (3rd vector). But the top of your head is pointing at a different direction (2nd vector). This changes your view. Refer to my article on lights and virtual cameras for more information (and pictures…).

So far, I haven’t used the 2nd vector to do much, so I’ve kept it at the default (0,1,0), which means point to the sky. Now for the problem…

Suppose you’re looking at something in front of you, say an (absolutely symmetrical) apple, and you move up while keeping the apple at the same position. You’re now looking down at it, aren’t you? Say you’re not the fidgety type, so your head is kept straight. Your head’s not pointed directly up to the sky, but it’s straight. You don’t know how, but you know it’s straight. This is important.

Now, slowly move towards the space directly above the apple. Your head is still kept “straight”. As in, if you tilted your head, the view changes substantially, and the apple appears “lopsided”. Here’s the problem (finally!). What happens when you look at the apple directly from above it?

Moving eye position while looking down

You can now tilt your head in any direction, and the apple still looks the same (as in you’re looking directly above it). Now the 2nd positional vector matters, because the view changes substantially (because the scene around the apple changes). The default “up” direction fails.

I can’t remember the name of this problem (or even if there was one). And I can’t find my OpenGL textbook that describes this, so I’m writing this from memory. If you can find me a reference to this, let me know.

So my young naive mind back then was “How about I don’t move the camera at all!” *sigh* So naive… As you can tell, my solution to move the entire world instead of moving the camera failed. It failed as in not solving the original problem I was trying to solve. But it worked flawlessly in that it worked exactly as the camera was made to move through the 3D world.

And I’ll tell you how I built that camera … next time.

Revolve the world around you

Sit or stand with your head pointing straight up. Tilt your head to your left. Note the view, the slant of the horizon, the movement (if any) of surrounding objects.

Tilt your head back to pointing straight up. Now imagine the view in front of you tilting to your right. Can you imagine the scene as having the same view as if you tilted your head left?

That was exactly what I was trying to achieve when I implemented a custom camera object to overcome a particular viewing problem in 3D. Well, you’ll have to wait for another article for the description of the problem. I’m going to just describe the function of that custom camera in this article.

So in 3D scenes, you have the scene objects and a virtual camera. Using the position and orientation of the camera, you can view the scene (by rendering the scene onto the viewing plane). This is analogous to the physical world.

Now, in the physical world, the scene, the set, and the props typically don’t move. Only the camera moves (we’ll leave out the human actors). I’m referring to the movement where an object goes from one place to another. Movement such as water flowing, or explosions aren’t included (as far as the discussion goes).

For a physical camera, there are limits. You can’t quite fly a camera through an explosion. You need special cameras to go through tiny openings. You’ve got to be careful when working with mirrors, because the camera (and cameraman) can be inadvertently captured (unless that was the effect). And you definitely can’t pass through walls.

A virtual camera in a 3D scene has none of those limitations. As far as the renderer is concerned, a camera is just a point, unless it’s modelled and treated as a 3D object. It can film the hottest of volcanic films, or be submerged in the depths of the seas, and remain undamaged. Now, the virtual camera might be limitless, but that’s not the point. Due to the transformations such as translations, rotations and scaling, the 3D scene itself can be modified at will.

I was inspired by a remark made by my university lecturer. He said that moving the camera towards a stationary object, is the same as moving the object towards the stationary camera. This also implied that rotating the camera clockwise around a stationary object, is the same as rotating the object anti-clockwise around the stationary camera.

This opened my eyes to another corollary. You don’t need to move the camera at all! You can move the entire scene instead.

So I set out to design a camera object where the entire 3D scene depended on it. What I mean is, instead of setting camera properties and have them work nicely with the 3D scene, the 3D scene itself conforms to the camera properties.

For example, if I set the camera position at (1,0,0), in actuality, the camera is still at (0,0,0). But the entire 3D scene is translated (-1,0,0).

What I did was set the camera at a default position, say (0,0,5) (I’m using the upright Y-axis), and set the camera’s “up” vector to (0,1,0) (meaning it’s head is pointing upwards, so it’s level with the ground). Then everything else is done with respect to this default camera orientation.

So why am I doing all this? I was bored, I had time then, and I wanted to solve a particular problem. I’ll tell you more about the mechanics of the camera, and the problem some other time…