We should reframe the question in a way that can better shed light on it.

Why was there such an impetus to undertake long, unpleasant voyages and forcibly transport such an obscene number of men, women and children across the Atlantic ocean, many of them to certain death? If we know why, then it's easier to explain why not to Europe.

Origin and Entrenchment of the Atlantic Slave Trade

Philip D. Curtin suggests in "Sugar Comes to the Atlantic Islands" that the islands of the Canaries for Spain and Madeira for Portugal served as laboratories for the slave trade, the first inklings of what was to come for Atlantic slavery. Essential to creating the slave trade was sugarcane; it was extremely labor-intensive. Although it had been grown on plantations in the Mediterranean, the climate was not amiable to year-round sugar harvests, which lowered productivity, and the conditions were less deadly for slaves, meaning that there was less continuous demand for new labor to replace losses (although thinking of it as a death camp is erroneous).

The Atlantic islands represent the first prototypes of a very different kind of plantation. Originally establishing "manorial" (as Curtin calls it) relations, there was awareness of the possibility of using Africans as slaves, and this likely motivated a turn towards importing slave labor. High death rates in terrible conditions meant the beginning of a continuous demand. By 1500 a notable, Fernao De Mello, was provided the island of São Tomé as a fief, and he established plantations there.

Demand for sugar rose in Europe, and so there was an urge to find as many "sugar islands" as they could. I must emphasize the small-scale of this production, however. Madeira at its peak produced 2,400 metric tons of sugar.

This is nothing compared to what came next. Early Spanish attempts to establish sugar colonies on Hispaniola peaked around 1,000 metric tons, hardly impressive by any means. What seems interesting here is that early on, sugar production was still very low, and most Spanish attempts failed. Why was this the case? Portugal had much stronger control over the slave trade at this early stage, making it far harder for Spanish slavers to purchase from the African coast.

Then, Brazil. By 1630 it was producing between 20,000 and 30,000 metric tons of sugar. It is at this point that we see escalation; Brazil emerged as a major sugar producer around the same time as the Caribbean.

António de Almeida Mendes, as a caveat, contends in "The Foundations of the System: A Reassessment of the Slave Trade to the Spanish Americas in the Sixteenth and Seventeenth Centuries” that we must also imagine these early journeys as experimental. Both Spaniards and Portuguese expended considerable effort in the early slave trade, and their expertise was transferred. We must thus imagine the slave trade as an enterprise; slave traders got better at it as time went on. Ships got bigger, mortality rates decreased, and shorter distances were traversed.

So by the mid-17th century we have a catalyst for large-scale plantation (sugarcane) and a growing expertise (thanks to 150 years of Portuguese and Spanish practice).

What is the result?

The Explosion of the Slave Trade

"But the secrets of the work being not well understood, the Sugars they made were very inconsiderable, and little worth, for two or three years. But they finding their errours by their daily practice, began a little to mind; and by new directions from Brasil, sometimes by strangers, now and again by their own people, (who being covetous of the knowledge of a thing, which so much concerned them in their particulars, and for the general good of the whole Island)

[...] two thirds of the iland will [now] be fit for plantations of Sugar, which will make it one of the richest Spots of Earth under the Sun". -Richard Ligon on the transformation of Barbados into a sugar colony. All spelling errors are transcribed from the text.

Sugarcane is a plant that is difficult to work with. You cannot just sit down and plant some sugarcane; in order to extract the sugar you need a vast plantation infrastructure. It thus naturally lent itself to economies of scale; in the same account, Ligon speaks of the buying out of smaller plantations.

So who exactly were these men from Brazil who helped? Was it the Portuguese? No. The dreaded Dutchman. Not flying, but in flight.

From J.H Galloway's "The Role of the Dutch in the Early American Sugar Industry", we learn that following their failed colony in Brazil, the Dutch became major investors into sugar production (and the Netherlands was the sugar refining capital of Europe). They helped set up sugarcane on Barbados and throughout the Caribbean, providing the necessary start-up capital and expertise in many cases.

At the same time, the three-roller mill emerged. Much more effective than previous mills at extracting sugar from the cane, it may be compared to the cotton gin in terms of the efficiency gains it gave planters, and the results were similar.

Statistics on the slave trade are also valuable here, to put this "explosion" into numbers.

http://www.slavevoyages.org/assessment/estimates

As the Caribbean as a whole starts becoming a major sugar producer, we see a profound growth of the slave trade. The capital, the expertise, the technology, and the contacts were all there. The English supplanted the Portuguese as the main slave traders; Liverpool by itself had 558 slaving ships plying the seas from 1780-1790, and the four major French slaving ports had 873. Statistics are terrifying; in that single last decade of the Ancien Regime, France transported 270,000 slaves, with a 300% increase in slaves exported per year from 1780 to 1791, almost all to Saint Domingue.

What the Hell does This Have to Do with the Question?

Sugarcane was not the only cash crop of the slave trade. Cotton in the American South was popular, and coffee was a favorite in the highlands of Saint Domingue (it was actually the growing industry during the last decade of the colony, not sugarcane, which had peaked due to exhausted soil and lack of new lowlands to expand to). Rice was also popular in select areas like the South Carolina Tidewater, and there's probably a few other crops I'm missing.

All of these crops grow poorly enough in Europe. The hunt for better places to grow sugarcane was what transformed the Atlantic Islands (even if the process was not consciously pursued with Madeira and the Canaries at first). In all these areas, a lack of local labor stemming from demographic annihilation/decline of indigenous populations due to disease, overworking, and a combination of overworking and disease, along with the pre-existing ties of Portugal to the West African slave trading networks, established a precedent.

Since plantation labour, not domestic servitude (for which there was plenty of cheap surplus labor in Europe to use, rather than expensive-to-import Africans) was the main driving force behind much of the slave trade, there was little reason to export slaves to European locales.

It's also important to note that the slave trade was expensive. Many slave traders were ultimately unsuccessful and failed to make a profit. James Rogers, a Bristol slave trader in the 1780s and 1790s, was an example of that; he had a great deal of trouble competing with other traders and was forced to go into debt in order to pay for the slaves he was buying.

That speaks not only to the difficulty of running the slave trade, but the hunger and rapaciousness of sugar plantations for more manpower to replace those lost to the endless suffering harvesting sugar imposed on them.

Britain and France had also, by the eighteenth century, become integrated into the slave trade network as producers of finished goods like textiles to sell to Africa in order to accrue the capital needed to purchase slaves (and consumed sugar from the Caribbean). It would have made little sense for one rung of the network to attempt to co-opt another rung, even if sugarcane could grow in Europe by some magic.

The slave trade was thus expensive, tied heavily to the production of sugarcane and its refining into sugar for European consumption, and had depended on a long evolution that required a heavy capital investment and the depletion of local labour. Sugarcane didn't grow well in most of Europe, and where it did grow well, you could grow it better in the Caribbean, with readier access to a labour force, pre-established networks of trade and exchange, and a better harvest.

Sources

Philip D. Curtin The Rise and Fall of the Plantation Complex: Essays in Atlantic History ("Sugar Comes to the Atlantic Islands" is an essay incorporated into the Major Problems in Atlantic History anthology).

Richard Ligon, A True and Exact History of the Island of Barbadoes, 1673. You know he isn't lying because he says true and exact in the title.

António de Almeida Mendes, "The Foundations of the System: A Reassessment of the Slave Trade to the Spanish Americas in the Sixteenth and Seventeenth Centuries”.

J.H Galloway "The Role of the Dutch in the Early American Sugar Industry".

McDade, Katie. “Liverpool Slave Merchant Entrepreneurial Networks, 1725-1807”. Business History 53, no. 7 (2011): 1092-1109.

Viles, Perry. “The Slaving Interest in the Atlantic Ports, 1763-1792”. French Historical Studies 7, no. 4 (1972): 529-543.

http://www.slavevoyages.org/assessment/estimates Obviously. Created by scholars in the field through hard-work in trading documents that is no longer possible since reading cursive is a lost art and the field of history is doomed (:v), this is an imperative source for anyone who wants to look at estimates or precise numbers in the slave trade in the Atlantic.

Morgan, Kenneth. “James Rogers and the Bristol Slave Trade.” Historical Research 76, no. 192 (2003): 189-216.

A lot of that stems from the fact that much (though not all) slave labor was used for agricultural purposes in the form of cash crops like sugar, tobacco, or cotton. Many of these cash crops were labor intensive and often incredibly harmful to a person's health. It's also worth noting that these cash crops were unable to be grown in many Western European climates.

Britain was especially unequipped to handle such a large agricultural task due to both an increase in population that occurred over that time, as well as the Inclosure Acts that began in 1773 which began to shift land from being commonly held by the people who lived there to being privately owned. The inclosing of many of these lands turned what was once farmland into pastureland since herding requires less labor. This, in turn, led to a larger population able to work in factories once the Industrial Revolution was in full swing.

But just because the British couldn't farm themselves didn't mean they didn't have farmland. In having colonies in the Americas, which was much more active agriculturally than Britain itself was, they did profit off the slave trade itself even if they didn't have many slaves imported. Because of this, there was little reason for many of the ships carrying slaves to make a stop in Europe once they had their cargo. It was easier to sail across the Atlantic to their destinations rather than sail north, stop, and then sail west.

As a side request, could you provide the animated map if possible?