Let's formalize our approach starting with the assumption that we are situated in a coordinate system where the sphere resides at the origin and the z-axis represents the view direction. If we find ourselves outside of this coordinate system, converting the input data to align with it is relatively straightforward. After performing the necessary calculations, we can effortlessly revert back to the original coordinate system.
We possess a directional vector d
that indicates the desired center point of the sprite (referred to as xyzPolar
in your code snippet). Additionally, we have information regarding the width w
and height h
of the sprite, understanding that the width extends along the x-axis while the height extends along the y-axis (due to the nature of the view-aligned coordinate system).
For any arbitrary scalar offset t
, we can define the center of the sprite as t * d
. Subsequently, the points on the sprite are determined by the following set:
{ t * d + x * (w/2, 0, 0) + y * (0, h/2, 0) | -1 <= x <= 1, -1 <= y <= 1 }
The parameters x
and y
represent the positional coordinates on the sprite, where (-1, -1)
signifies the bottom left corner and (0, 0)
represents the center. Our specific interest lies in determining the point closest to the center of the sphere, featuring a distance of r
(equivalent to the sphere's radius). Hence:
min (t * dx + x * w/2)^2 + (t * dy + y * h/2)^2 + (t * dz)^2 = r^2
x, y in [-1, 1]
If we possess knowledge of the parameters x
and
y</code) pertaining to this closest point, we can effectively solve for <code>t</code, thereby obtaining the final position of the sprite's center.</p>
<p>However, in the absence of these parameters, let's dissect this formula:</p>
<pre><code>( min (t * dx + x * w/2)^2 ) + ( min (t * dy + y * h/2)^2 ) + (t * dz)^2 = r^2
x in [-1, 1] y in [-1, 1]
The first two terms are minimized if we set
x = -2 dx t / w
y = -2 dy t / h
In such instances, both terms would equate to zero, allowing us to derive t = r / abs(dz)
. Essentially, this will position the sprite on the xy-aligned plane where z = +- r
. This scenario holds true when dealing with an infinite sprite, without imposing restrictions on x
and y
.
Regrettably, we are not working with an infinite sprite. It becomes imperative to confine x
and
y</code) within the permissible range. Therefore, upon identifying a potential <code>t
, it is crucial to verify its validity by computing
x
and
y</code using the aforementioned formulas and ensuring their compliance with the allowable boundaries. Validation occurs if the nearest point exists within the sprite's central region as opposed to residing on an edge or corner.</p>
<p>Fortunately, the number of viable values for <code>x
and
y
that necessitate examination is limited. The algorithm orchestrates the calculation of
t
across all prospective values, validates the solution, and retains only the substantiated outcome. What possible values exist for
x
and
y
?
We are already familiar with scenarios under -1 <= x <= 1
and -1 <= y <= 1
. All values within this bracket remain equivalent since they nullify the initial two terms (thus having no bearing on the eventual result). Furthermore, alternative cases emerge for each variable, wherein either x = -1
or
x = 1</code) (likewise for <code>y</code). This gives rise to a compilation of nine permutations requiring resolution. Yet, optimization abounds. Given that <code>t
,
w</code), and <code>h
assume positive values,
x</code) and <code>y
exhibit an inverse association with
dx</code) and <code>dy</code) respectively. For instance, if <code>dx
proves positive, scrutinizing instances where
x = -1
suffices or leads to the vanishing of the initial term (effectively precluding the possibility of the sprite's right boundary serving as the closest point provided the direction vector extends rightward). Similarly, when
dx</code) or <code>dy</code) emulate exact zeroes, immediate deduction involves the erasure of the corresponding term, obviating the need to consider the counter case. In addition, should <code>dz = 0</code), bypass evaluation of situations where the inaugural pair of terms diminishes.</p>
<p>As a result, the process condenses down to four circumstances. A brief recap entails highlighting the quadratic equations attributable to the three distinct cases for each variable:</p>
<pre><code> first term
-1 = x dx^2 * t^2 - dx * t * w + w^2 / 4
-1 < x < 1 0
x = 1 dx^2 * t^2 + dx * t * w + w^2 / 4
second term
-1 = y dy^2 * t^2 - dy * t * h + h^2 / 4
-1 < y < 1 0
y = 1 dy^2 * t^2 + dy * t * h + h^2 / 4
Upon adjudicating the four cases warranting evaluation, expedite solving for
t</code), subsequently determine <code>x
and
y</code before confirming whether they align with the specific circumstance (e.g., if presented with <code>x = 1
, cross-check against
x >= 1
et cetera). Finally, compute the sprite center via
t * d
.
In essence, although not necessarily more elegant than existing methods, the elaborated approach ensures enhanced accuracy.