A system for rendering lesions on a geometric surface model of a geometric structure, the system comprising: an instrument for applying ablation to generate lesions; a computer-based model construction system configured to be coupled to a device that includes at least one sensor configured to acquire a set of original location data points to facilitate generating the geometric surface model, the computer-based model construction system further configured to: create a three-dimensional texture map which is a lattice of voxels each having an associated tissue necrosis value and large enough to include an entire bounding volume of the geometric surface model, wherein the 3D texture map is drawn onto a polygon comprising vertices having coordinates in 3D space in addition to one or more coordinates in the 3D texture map, which describe a portion of the texture map that gets drawn onto that polygon; increase a tissue necrosis value of each voxel in the 3D texture map that is within a given distance to an ablation location by an amount that is an incrementing function of a value of at least one parameter associated with an ablation procedure at the ablation location to generate a total tissue necrosis value for each voxel indicating a level of necrosis during ablation, wherein the value of the at least one parameter is acquired by using the instrument; render in real-time at least one lesion on the geometric surface model by drawing triangles of the geometric surface model using each geometric vertex of the triangles as index into the 3D texture map based on the total tissue necrosis values; and display the geometric surface model and said at least one rendered lesion. an instrument for applying ablation to generate lesions; a computer-based model construction system configured to be coupled to a device that includes at least one sensor configured to acquire a set of original location data points to facilitate generating the geometric surface model, the computer-based model construction system further configured to: create a three-dimensional texture map which is a lattice of voxels each having an associated tissue necrosis value and large enough to include an entire bounding volume of the geometric surface model, wherein the 3D texture map is drawn onto a polygon comprising vertices having coordinates in 3D space in addition to one or more coordinates in the 3D texture map, which describe a portion of the texture map that gets drawn onto that polygon; increase a tissue necrosis value of each voxel in the 3D texture map that is within a given distance to an ablation location by an amount that is an incrementing function of a value of at least one parameter associated with an ablation procedure at the ablation location to generate a total tissue necrosis value for each voxel indicating a level of necrosis during ablation, wherein the value of the at least one parameter is acquired by using the instrument; render in real-time at least one lesion on the geometric surface model by drawing triangles of the geometric surface model using each geometric vertex of the triangles as index into the 3D texture map based on the total tissue necrosis values; and display the geometric surface model and said at least one rendered lesion. create a three-dimensional texture map which is a lattice of voxels each having an associated tissue necrosis value and large enough to include an entire bounding volume of the geometric surface model, wherein the 3D texture map is drawn onto a polygon comprising vertices having coordinates in 3D space in addition to one or more coordinates in the 3D texture map, which describe a portion of the texture map that gets drawn onto that polygon; increase a tissue necrosis value of each voxel in the 3D texture map that is within a given distance to an ablation location by an amount that is an incrementing function of a value of at least one parameter associated with an ablation procedure at the ablation location to generate a total tissue necrosis value for each voxel indicating a level of necrosis during ablation, wherein the value of the at least one parameter is acquired by using the instrument; render in real-time at least one lesion on the geometric surface model by drawing triangles of the geometric surface model using each geometric vertex of the triangles as index into the 3D texture map based on the total tissue necrosis values; and display the geometric surface model and said at least one rendered lesion.