The emergence of custom tapestry printing companies over the past few years has seen a remarkable shift in the dynamics of the textile industry. It's a development that testifies to the innovative, versatile nature of the industry, and its ability to adeptly respond to customer demand for personalized products. As we analyse the trajectory of this industry, we can make some predictions about the trends that will shape its future.

Tapestry has a rich and storied history, dating back to the Hellenistic times. Traditionally woven by hand, it was a labor-intensive, time-consuming process that required high levels of skill. However, the advent of technology has dramatically changed the landscape, enabling the production of custom tapestry on a scale previously unimaginable.

One of the key trends influencing the future of custom tapestry printing companies is the continual advancement of digital printing technologies. Using a method analogous to traditional inkjet printing, digital textile printers can produce intricate designs with a wide array of colors directly onto fabric.

Yet, this isn't merely a matter of transferring a design from a computer screen to a physical medium. The complex interplay between the ink, fabric and heat used in the printing process can significantly affect the final product's color accuracy, durability, and feel. This necessitates a deep understanding of colorimetry (the science of creating desired colors), rheology (the study of the flow and deformation of matter), and thermodynamics.

One of the immediate benefits of digital printing is the reduction in the lead time for producing custom tapestry. The traditional tapestry production process involved a time-consuming setup, which made it economically unviable for producing small quantities or individual pieces. Digital printing, however, allows for on-demand production, enabling companies to swiftly respond to bespoke orders.

The ability to produce custom tapestry rapidly has significant implications for the market dynamics. It allows for a business model based on 'fast fashion', where designs can be quickly changed to align with the latest trends or customer preferences. This agility can lend a competitive edge to the companies that can leverage it effectively.

However, this model isn't without its trade-offs. Fast fashion has been heavily criticized for its environmental impact, particularly in terms of waste generation and resource consumption. Adopting a fast-fashion model, therefore, presents a challenge for tapestry printing companies to balance profitability with responsible, sustainable practices.

In the quest for sustainability, companies are exploring various eco-friendly alternatives. One such alternative is the use of organic inks, which are devoid of harmful chemicals. The use of such inks is often accompanied by a shift towards more sustainable fabrics, like organic cotton or recycled polyester. This, however, may lead to a slight increase in production costs, which companies have to balance with the consumers' willingness to pay for environmentally friendly products.

Simultaneously, technologies like Augmented Reality (AR) could shape the future of this industry. AR can enable customers to visualize how a specific custom tapestry would look in their space before making a purchase, thereby enhancing the buying experience and reducing the likelihood of returns.

In conclusion, the future of custom tapestry printing companies will likely be shaped by advancements in digital printing technologies, the ability to respond swiftly to market trends, and the growing emphasis on sustainability. It's a landscape that demands a constant balance between innovation, profitability, and responsibility. It is also a landscape ripe with opportunities for those who can navigate it adeptly.