Oncogene amplification, a major driver of cancer pathogenicity, is often mediated through focal amplification of genomic segments. Recent results implicate extrachromosomal DNA (ecDNA) as the primary driver of focal copy number amplification (fCNA) - enabling gene amplification, rapid tumor evolution, and the rewiring of regulatory circuitry. Resolving an fCNA's structure is a first step in deciphering the mechanisms of its genesis and the fCNA's subsequent biological consequences. We introduce a computational method, AmpliconReconstructor (AR), for integrating optical mapping (OM) of long DNA fragments (>150 kb) with next-generation sequencing (NGS) to resolve fCNAs at single-nucleotide resolution. AR uses an NGS-derived breakpoint graph alongside OM scaffolds to produce high-fidelity reconstructions. After validating its performance through multiple simulation strategies, AR reconstructed fCNAs in seven cancer cell lines to reveal the complex architecture of ecDNA, a breakage-fusion-bridge and other complex rearrangements. By reconstructing the rearrangement signatures associated with an fCNA's generative mechanism, AR enables a more thorough understanding of the origins of fCNAs.