Please explain compute_aabb

Hi, thank you for providing the code and for your great paper!
Could you please explain the compute_aabb function to me?

diff-surfel-rasterization/cuda_rasterizer/forward.cu

Lines 122 to 148 in a97d67a

    
           __device__ bool compute_aabb( 
        
           	glm::mat3 T,  
        
           	float2& point_image, 
        
           	float2 & extent 
        
           ) { 
        
           	float3 T0 = {T[0][0], T[0][1], T[0][2]}; 
        
           	float3 T1 = {T[1][0], T[1][1], T[1][2]}; 
        
           	float3 T3 = {T[2][0], T[2][1], T[2][2]}; 
        
           	// Compute AABB 
        
           	float3 temp_point = {1.0f, 1.0f, -1.0f}; 
        
           	float distance = sumf3(T3 * T3 * temp_point); 
        
           	float3 f = (1 / distance) * temp_point; 
        
           	if (distance == 0.0) return false; 
        
           	point_image = { 
        
           		sumf3(f * T0 * T3), 
        
           		sumf3(f * T1 * T3) 
        
           	};   
        
           	float2 temp = { 
        
           		sumf3(f * T0 * T0), 
        
           		sumf3(f * T1 * T1) 
        
           	}; 
        
           	float2 half_extend = point_image * point_image - temp; 
        
           	extent = sqrtf2(maxf2(1e-4, half_extend)); 
        
           	return true;

I can't quite understand how it is derived.

I can't understand this part either. If you understand, please let me know. Thank you @AlexRoss-WHS

Thank you for answering my question perfectly @hbb1

Thank you @hbb1 for taking the time for such a detailed answer! Now that I understand the code and maths I think there are two small typos in your answer:

It sould be $b = -2 * \text{sum}(f T_0 T_3)$
and ${x_1 + x_2 \over 2} ={ -b \over 2 a}$

Hi, everyone. @CanCanZeng @AlexRoss-WHS @FantasticOven2
In case you are not aware of it, there is a bug in the current AABB computation. We should directly calculate 3 sigma AABB in the object space instead of calculating 1 sigma AABB and multiplying it with 3 in the screen space. The reason is here #15

	__device__ bool compute_aabb(
	glm::mat3 T,
	float2& point_image,
	float2 & extent
	) {
	float3 T0 = {T[0][0], T[0][1], T[0][2]};
	float3 T1 = {T[1][0], T[1][1], T[1][2]};
	float3 T3 = {T[2][0], T[2][1], T[2][2]};

	// Compute AABB
	float3 temp_point = {1.0f, 1.0f, -1.0f};
	float distance = sumf3(T3 * T3 * temp_point);
	float3 f = (1 / distance) * temp_point;
	if (distance == 0.0) return false;

	point_image = {
	sumf3(f * T0 * T3),
	sumf3(f * T1 * T3)
	};

	float2 temp = {
	sumf3(f * T0 * T0),
	sumf3(f * T1 * T1)
	};
	float2 half_extend = point_image * point_image - temp;
	extent = sqrtf2(maxf2(1e-4, half_extend));
	return true;