Ceres-Solver cost function inheritance error: Templates may not be virtual

I have been using ceres-solver for a long time now and it is an amazing tool. My usage up until now was not based on reusable code and I am trying to improve this. Ceres uses a specific structure with a specific templated method as an interface to its automatic differentiation. In the problem that I am trying to solve, inheritance makes sense because the different cost functions that I need to have are very similar to each other. I have created an example that is similar (but it makes no sense, sorry). Imagine we want to be able to find the polygon that has a given area. In my example, polygons can be either triangles or rectangles. With this in mind, it makes sense to have a base class that implements everything and specific classes that implement, in this case, the area computation for each specific polygon:

ShapeCostFunction

class shapeAreaCostFunction
{
public:
    shapeAreaCostFunction(double desired_area): desired_area_(desired_area){}

    template<typename T>
    bool operator()(const T* shape, T* residual) const{
        residual[0] = T(desired_area_) - area(shape);
        return true;
    }

    template<typename T>
    virtual T area(const T* shape) const = 0;

protected:
    double desired_area_;
};

RectangleCostFunction

#include "shapeAreaCostFunction.h"
#include "areaLibrary.h"

class rectangleAreaCostFunction : public shapeAreaCostFunction
{
public:
    rectangleAreaCostFunction(double desired_area): shapeAreaCostFunction(desired_area){}

    template<typename T>
    T area(const T* triangle) const
    {
        return rectangleArea(triangle);
    }
};

TriangleCostFunction

#include "shapeAreaCostFunction.h"
#include "areaLibrary.h"

class triangleAreaCostFunction : public shapeAreaCostFunction
{
public:
    triangleAreaCostFunction(double desired_area): shapeAreaCostFunction(desired_area){}

    template<typename T>
    T area(const T* triangle) const
    {
        return triangleArea(triangle);
    }
};

AreaLibrary

template<typename T>
T rectangleArea(const T* rectangle)
{
    return rectangle[0]*rectangle[1];
}

template<typename T>
T triangleArea(const T* triangle)
{
    return rectangleArea(triangle)/T(2);
}

Main

#include <ceres/ceres.h>
#include <iostream>

#include "rectangleAreaCostFunction.h"
#include "triangleAreaCostFunction.h"
#include "areaLibrary.h"

int main(int argc, char** argv){

    // Initialize glogging
    //google::InitGoogleLogging(argv[0]);

    // Get values
    /// Get total area
    double total_area;
    std::cout<<"Enter the desired area: ";
    std::cin>>total_area;
    /// Get initial rectangle
    double rect[2];
    std::cout<<"Enter initial rectangle base: ";
    std::cin>>rect[0];
    std::cout<<"Enter initial rectangle height: ";
    std::cin>>rect[1];
    /// Get initial triagnle
    double tri[2];
    std::cout<<"Enter initial triangle base: ";
    std::cin>>tri[0];
    std::cout<<"Enter initial triangle height: ";
    std::cin>>tri[1];

    // Copy initial values
    double rect_ini[] = {rect[0],rect[1]};
    double tri_ini[] = {tri[0],tri[1]};

    // Create problem
    ceres::Problem problem;
    ceres::CostFunction* cost_function_rectangle = new ceres::AutoDiffCostFunction<rectangleAreaCostFunction, 1, 2>(
            new rectangleAreaCostFunction(total_area));
    ceres::CostFunction* cost_function_triangle = new ceres::AutoDiffCostFunction<triangleAreaCostFunction, 1, 2>(
            new triangleAreaCostFunction(total_area));
    problem.AddResidualBlock(cost_function_rectangle, NULL, rect);
    problem.AddResidualBlock(cost_function_triangle, NULL, tri);

    // Solve
    ceres::Solver::Options options;
    options.linear_solver_type = ceres::DENSE_QR;
    options.minimizer_progress_to_stdout = true;
    options.max_num_iterations = 10;
    ceres::Solver::Summary summary;
    ceres::Solve(options, &problem, &summary);

    // Compute final areas
    double rect_area = rectangleArea(rect);
    double tri_area = triangleArea(tri);

    // Display results
    std::cout << summary.FullReport() << std::endl;
    std::cout<<"Rectangle: ("<<rect_ini[0]<<","<<rect_ini[1]<<") -> ("<<rect[0]<<","<<rect[1]<<") total area: "<<rect_area<<"("<< rect_area - total_area<<")"<<std::endl;
    std::cout<<"Triangle: ("<<tri_ini[0]<<","<<tri_ini[1]<<") -> ("<<tri[0]<<","<<tri[1]<<") total area: "<<tri_area<<"("<< tri_area - total_area<<")"<<std::endl;

    // Exit
    return 0;
}

The problem with this is that templated functions cannot be virtual as explained several times in stackoverflow (here and here). However, it seems there are some workarounds using boost::any. I have tried to use this in my example with no success. I have also tried to move the template from the class method to the class, similar to here, but ceres do not accept it as a cost function.

My questions are (and please, keep in mind that I am restricted to have the method template<typename T> bool operator()(...)const otherwise I cannot interact with ceres):

Does it make sense to have an inheritance system like this (imagine this is a much complex problem than the example)?
Is there a way to keep this inheritance system and make the code work or shall I just move everything to separate functions and call the right function from each template<typename T> bool operator()(...)const class method?

Thank you in advance.

Solution

I can think of two approaches.

First, compose lambdas. Second, use CRTP.

Compose Lambdas

This is best done with c++14.

template<class Area>
auto cost_function(Area area, double desired){
  return [=](auto const* shape, auto* residual){
    using T=std::decay_t<decltype(*shape)>;
    residual[0] = T(desired_area_) - area(shape);
    return true;
  };
}
auto triangle = [](auto* shape){return triangleArea(shape);};

To create a triangle area cost function:

auto tri_cost = cost_function(triangle, 3.14159);

and to get the type, decltype(tri_cost).

So:

auto tri_cost = cost_function(triangle, 3.14159);
ceres::CostFunction* cost_function_triangle = new ceres::AutoDiffCostFunction<decltype(tri_cost), 1, 2>(
        new decltype(tri_cost)(tri_cost));

You can do a similar composition technique without lambdas, but it is more tedious. You can also wrap some of these naked new's up in helper functions.

CRTP

template<class D>
class shapeAreaCostFunction {
public:
  shapeAreaCostFunction(double desired_area): desired_area_(desired_area){}

  template<typename T>
  bool operator()(const T* shape, T* residual) const{
    residual[0] = T(desired_area_) - static_cast<D const*>(this)->area(shape);
    return true;
  }
protected:
  double desired_area_;
};

modify derived types like this:

class triangleAreaCostFunction :
  public shapeAreaCostFunction<triangleAreaCostFunction>
{
  using base=shapeAreaCostFunction<triangleAreaCostFunction>;
public:
  triangleAreaCostFunction(double desired_area): base(desired_area){}

  template<typename T>
  T area(const T* triangle) const
  {
    return triangleArea(triangle);
  }
};

this is known as using the CRTP to implement static polymorphism.