Dynamically recalculating normals after vertex displacement - unity3d

Question

Can anyone let me know if I'm on the right tack with this: I have a vertex shader that bumps outward dynamically depending on a point passed in (think a mouse running under a rug). In order for the lighting to update properly, I need to recalculate the normals after modifying the vertex position. I have access to each vertex point as well as the origin.
My current thinking is I do some sort of math to determine the tangent / bitangent and use a cross product to determine the normal. My math skills aren't great, what would I need to do to determine those vectors?
Here's my current vert shader:
void vert(inout appdata_full v)
{
float3 worldPos = mul(unity_ObjectToWorld, v.vertex).xyz;
float distanceToLift = distance(worldPos, _LiftOrigin);
v.vertex.y = smoothstep(_LiftHeight, 0, distanceToLift / _LiftRadius) * 5;
}

Answer 1

A simple solution is covered in this tutorial by Ronja, which I'll summarize here with modifications which reflect your specific case.
First, find two points offset from your current point by a small amount of tangent and bitangent (which you can calculate from normal and tangent):
float3 posPlusTangent = v.vertex + v.tangent * 0.01;
worldPos = mul(unity_ObjectToWorld, posPlusTangent).xyz;
distanceToLift = distance(worldPos, _LiftOrigin);
posPlusTangent.y = smoothstep(_LiftHeight, 0, distanceToLift / _LiftRadius) * 5;
float3 bitangent = cross(v.normal, v.tangent);
float3 posPlusBitangent = v.vertex + bitangent * 0.01;
worldPos = mul(unity_ObjectToWorld, bitangent).xyz;
distanceToLift = distance(worldPos, _LiftOrigin);
posPlusBitangent.y = smoothstep(_LiftHeight, 0, distanceToLift / _LiftRadius) * 5;
Then, find the difference between these offsets and the new vertex pos to find the new tangent and bitangent, then do another cross product to find the resulting normal:
float3 modifiedTangent = posPlusTangent - v.vertex;
float3 modifiedBitangent = posPlusBitangent - v.vertex;
float3 modifiedNormal = cross(modifiedTangent, modifiedBitangent);
v.normal = normalize(modifiedNormal);
Altogether:
float find_offset(float3 localV)
{
float3 worldPos = mul(unity_ObjectToWorld, localV).xyz;
float distanceToLift = distance(worldPos, _LiftOrigin);
return smoothstep(_LiftHeight, 0, distanceToLift / _LiftRadius) * 5;
}
void vert(inout appdata_full v)
{
v.vertex.y = find_offset(v.vertex);
float3 posPlusTangent = v.vertex + v.tangent * 0.01;
posPlusTangent.y = find_offset(posPlusTangent);
float3 bitangent = cross(v.normal, v.tangent);
float3 posPlusBitangent = v.vertex + bitangent * 0.01;
posPlusTangent.y = find_offset(posPlusBitangent);
float3 modifiedTangent = posPlusTangent - v.vertex;
float3 modifiedBitangent = posPlusBitangent - v.vertex;
float3 modifiedNormal = cross(modifiedTangent, modifiedBitangent);
v.normal = normalize(modifiedNormal);
}
This is a method of approximation, but it may be good enough!